MATHEMATICAL PAPERS OF THE LATE
GEORGE GREEN. *
CTamfcrftgc:
PBINTED BY C. J. CLAY, M.A. AT THE UNIVEBSITY PBESS.
MATHEMATICAL PAPERS OF THE LATE
GEORGE GREEN, It
FELLOW OF GONVILLE AND CAIUS COLLEGE, CAMBBIDGE.
EDITED BY N.
M.
FERRERS,
M.A.,
FELLOW AND TUTOE OF GONVILLE AND CAIUS COLLEGE.
\*
Uonfcon
:
MACMILLAN AND 1871. [All Rights resewed.]
CO.
PREFACE. HAVING been
requested
by the
and
Master
Fellows
of
Gonville and Caius College to superintend an edition of the
mathematical writings of the late George Green, I have the task to the best of
my
ability.
The
publication
fulfilled
may
be
opportune at present, as several of the subjects with which they are directly
or
indirectly
concerned, have recently been in-
troduced into the course of mathematical study at Cambridge.
They have
also
an interest as being the work of an almost
entirely self-taught mathematical genius.
George Green was born at Sneinton, near Nottingham, in 1793.
He commenced
lege, in October, 1833,
residence at Gonville and Caius Col-
and in January, 1837, took
of Bachelor of Arts as Fourth Wrangler.
It is hardly neces-
sary to say that this position, distinguished as
inadequately represented his
his degree
mathematical power.
it
was, most
He
laboured
under the double disadvantage of advanced age, and of inability to submit entirely to the course of systematic training needed for the highest places
He
in the Tripos.
was elected
to a
fellowship of his College in 1839, but did not long enjoy this position, as
he died in 1841.
pages will sufficiently
The contents
shew the heavy
loss
world sustained by his premature death.
of the following
which the
scientific
PREFACE.
VI
A slight
sketch of the papers comprised in this volume
may
not be uninteresting.
The
paper, which
first
also the longest
is
and perhaps the
most important, was published by subscription at Nottingham in 1828. It was in this paper that the term potential was first introduced to denote the result obtained by adding together the masses of
all
the particles of a system, each divided by
distance from a given point.
In this essay, which
is
its
divided
into three parts, the properties of this function are first con-
and they are then applied, in the second and third to the theories of magnetism and electricity respectively.
sidered, parts,
The
essay which the author has given in
full analysis of this
his Preface, renders
necessary.
portions
any detailed description in
In connexion with
Thomson and
of
this essay,
Tait's
this place
un-
the corresponding
Natural Philosophy should
be studied, especially Appendix A. to Chap.
I.,
and Arts. 482
550, inclusive.
The next
"
paper,
On
analogous to the Electric Philosophical
fluid
the n
power of the
is
it,
"On
;
was
laid before the
Cambridge
Edward Ffrench Bromhead,
in
taken to be inversely proportional to
distance.
great analytical power, interesting
Sir
of the Equilibrium of Fluids
of repulsion of the particles of the supposed
here considered ih
Laws
Fluid,'*
Society by
The law
1832.
the
is
This paper, though displaying
perhaps rather curious than practically
and a similar remark applies
to that
which succeeds
the determination of the attractions of Ellipsoids of
variable Densities," which, like its predecessor,
was communi-
cated to the Cambridge Philosophical Society by Sir E. F.
Bromhead.
Space of n dimensions
is
here considered, and
the surfaces of the attracting bodies are supposed to be repre-
PREFACE.
Vii
sented by equations formed by equating to unity the sums of the squares of the n variables, each divided by an appropriate It is of course possible to adapt the
coefficient.
paper to the case of nature
The next
paper,
"On
by supposing n
the Motion of
formula of this
= 3.
Waves
canal of small depth and width," though short,
in a variable is
interesting.
was read before the Cambridge Philosophical Society, on May 15, 1837, and a Supplement to it on Feb. 18, 1839.
It
On
Dec. 11, 1837, were communicated two of his most valuable
memoirs, "
On
"On
the Reflexion and Refraction of Sound," and
of
surface
two non-crystallized media."
should be studied together. is,
a
common
the Reflexion and Refraction of Light at the
The question
in fact, that of the propagation of fluid.
These two papers discussed in the
normal vibrations through
Particular attention should be paid to the
which, from the differential equations of motion,
Optics as
of a
to
exceeds the critical angle.
By
mode
in
deduced an
is
that
phenomenon analogous Total internal reflection when the angle
explanation
first
known
in
of incidence
supposing that there are pro-
pagated, in the second medium, vibrations which rapidly diminish in intensity, and
become evanescent at
sensible distances,
the change of phase which accompanies this phenomenon clearly
is
brought into view.
The immediate
object of the next paper,
"
On
the Reflexion
and Refraction of Light at the common surface of two nonof light what in the crystalline media," is to do for the theory former paper has been done for that of sound.
a manner which will present
mastered the former paper.
This
little difficulty to
But
extending far beyond this subject.
this
is
done in
one who has
paper has an interest
For the purpose of explain-
PREFACE.
Vlll
the
of
transversal vibrations through the becomes ether, necessary to investigate the equations of motion of an elastic solid. It is here that Green
ing
propagation
luminiferous
for the first
it
time enunciates the principle of the Conservation
o work, which he bases on the assumption of the impossibility
This principle he enunciates in the
of a perpetual motion.
"In whatever manner the elements of any
following words:
material system
upon each
act
may
other, if all the internal
be multiplied by the elements of their respective direc-
forces
sum
any assigned portion of the mass will always be the exact differential of some function." This function, it will be seen, is what is now known under the name of
tions,
the total
for
Potential Energy, and the above principle to stating that the
of the system
sum
of the Kinetic
and Potential Energies
This function, supposing the dis-
constant.
is
in fact equivalent
is
placements so small that powers above the second is
neglected,
medium
shewn
for the
may be
most general constitution of the
to involve twenty-one coefficients,
which reduce to nine
rectangular
medium symmetrical with respect to three planes, to five in the case of a medium symmetrical
around an
axis,
in the case of a
crystallized
and
two in the case of an
to
medium.
The present paper
is
isotropic or
un-
devoted to the
consideration of the propagation of vibrations from one of two
media of
this nature.
called respectively
A
The two
coefficients
above mentioned,
and B, are shewn to be proportional to
the squares of the velocities of propagation of normal and transversal vibrations respectively.
the statical interpretation
shewn that
(see
Thomson and
A-%B
is
It is to
not also given.
Tait's
be regretted that
It
may however be
Natural Philosophy,
measures the resistance of the
p.
medium
711
(m.))
to
com-
PREFACE. pression or dilatation, or
IX
its elasticity
sures its resistance to distortion, or its rigidity. of the
medium,
it
may be
The equilibrium
shewn, cannot be stable, unless both
A
of these quantities are positive*.
Supplement
supplying certain omissions, immediately follows
In the next paper, talline
"
On
assumed as a starting-point and applied description.
to this paper
it.
the Propagation of Light in Crys-
Media," the principle of Conservation of
It is first
B mea-
of volume, while
to a
Work
is
medium
assumed that the medium
is
again
of any
symmetrical
with respect to three planes at right angles to one another, by
which supposition the twenty-one
coefficients previously
men-
Fresnel's supposition, that the
tioned are reduced to nine.
vibrations affecting the eye are accurately in front of the wave, is
then introduced, and a complete explanation of the phe-
nomena
of polarization
that the vibrations
is
to follow,
on the hypothesis
constituting a plane-polarized ray are in
the plane of polarization.
former paper
shewn
The hypothesis adopted
in
the
that these vibrations are perpendicular to the
plane of polarization
is
then resumed, and an explanation
not of
by the aid of a subsidiary assumption unfortunately the same simple character as those previously intro-
duced
that for the three principal waves the wave-velocity
arrived at,
depends on the direction of the disturbance only, and
dependent of the position of the wave's
front.
is
in-
The paper
concludes by taking the case of a perfectly general medium,
and
it
is
shewn that
Fresnel's
supposition of the vibrations
being accurately in the wave-front, gives rise to fourteen re*
In comparing Green's paper with the passage in Thomson and Tait's Natural Philosophy above referred to, it should be remarked that the A of the former is equal to the m - \n of the latter, and that B=n.
X
PREFACE.
among the twenty-one coefficients, which virtually reduce the medium to one symmetrical with respect to three
lations
planes at right angles to one another.
This paper, read Another,
"On
May
20, 1839,
was his
last production.
the Vibrations of Pendulums in Fluid Media,"
read before the Royal Society of Edinburgh, on Dec. 16, 1833, will
be found at the end of
considered
is
this collection.
The problem here
that of the motion of an inelastic fluid agitated
by the small vibrations
of a solid ellipsoid,
moving
parallel to
itself.
I have to express
my
thanks to the Council of the Cam-
bridge Philosophical Society, and to that of the Royal Society of Edinburgh, for the permission to reproduce the papers published in their respective Transactions which they have kindly given.
N. M.
GONVILLE AND CAIUS COLLEGE, Dec. 1870.
FERRERS.
CONTENTS. PAGE
An
Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism
Preface
...
....
Introductory Observations
General Preliminary Results
1
...
...
...
...
...
...
...
...
...
...
...
9
...
19
...
...
...
.3
Application of the preceding results to the Theory of Electricity
;.
...
...
42
Application of the preliminary results to the Theory of Mag-
netism
...
...
...
...
...
...
...
83
Mathematical Investigations concerning the Laws of the Equilibrium of Fluids analogous to the Electric Fluid, with other similar researches
On
...
...
...
...
...
the Determination of the Exterior and Interior Attractions
185
of Ellipsoids of Variable Densities
On
the Motion of
Waves
in a variable canal of small depth
223
and width
On
the Reflexion and Refraction of Sound
On
the
Laws
common
231
...
of the Reflexion and Refraction of Light at the surface of
two non-crystallized Media
Note on the Motion of Waves Supplement to a
...
...
in Canals
Memoir on the Reflexion and Refraction
the Propagation of Light in crystallized
281
Media
...
Researches on the Vibration of Pendulums in Fluid Media
APPENDIX
243 271
of Light
On
117
...
291
...
313 325
ERRATA. Page
23, line 11, for there read these.
dzdy read dydz.
25, for
,,
23,
"
29>
-
"
29 >
"
,,
36,
7>
**% read ^-
**-
22, after co-ordinates, insert
37,
,,
of.
2 from bottom, for dV, read 5'V.
for axes, read
axis.
,,
43,
,,
8,
,,
46,
,,
19, after radius, insert
53,
,,
7,
for p-read
is.
(g).
3
54,
read 27r/ 3 s 16, for 47ra/ read 4iraf
56,
19, for
11, for 47r/
54,
.
2
60,
13,fcrJ^
64,
,,
71,
4 from bottom, before a potential insert throughout for dw and dw read dw. for
dw read
dt7.
72,
,,
18,
74,
,,
24, /or his read this.
84,
,,
11, for
,,
3 2 20, for r read r 17, for sin 0' read sin
{J(i)
Z7(2)
88,
read
-^
-^
.
.
89,
18,
89,
90,
,,
92
2
U
for
/or
24, for
read
0.
tf(D.
^ rga^^read |TT
^_.
2
d' d)
107
.
,
20,' 'for r
2 -
dx2
=
read r 2
for these read thus.
123
,,
19,
129
,,
24, for sin
-GT
d2 -=-5 + dx2
of.
AN ESSAY ON THE APPLICATION OF MATHEMATICAL ANALYSIS TO THE THEORIES
OF ELECTRICITY AND MAGNETISM.
Published at Nottingham, in 1828.
1
PREFACE. AFTER
had composed the following Essay, I naturally felt. anxious to become acquainted with what had been effected by former writers on the same subject, and, had it been practicable, I should have been glad to have given, in this place, an hisI
torical sketch of its progress;
my
limited sources of information,
however, will by no means permit me to do so ; but probably I may here be allowed to make one or two observations on the
way, more particularly as an will thus offer of itself, noticing an excellent paper, opportunity to the Royal Society by one of the most illustrious presented few works which have fallen in
my
members of that learned body, which appears little
to have attracted be found not but on will attention, which, examination,
unworthy the man who was able to lay the foundations of pneumatic chymistry, and to discover that water, far from being according to the opinions then received, an elementary substance, was a compound of two of the most important gases in nature. It is almost needless to say the author just alluded to is the CAVENDISH, who, having confined himself to such
celebrated
simple methods, as may readily be understood by any one possessed of an elementary knowledge of geometry and fluxions, has rendered his paper accessible to a great number of readers;
and although, from subsequent remarks, he appears dissatisfied with an hypothesis which enabled him to draw some important conclusions, it will readily be perceived, on an attentive perusal of his paper, that a trifling alteration will suffice to render the
whole perfectly legitimate*. *
In order to
make this
CAVENDISH'S proposiand examine with some attention the method
quite clear, let us select one of
tions, the twentieth for instance,
12
4
PREFACE. Little appears to
have been effected in the mathematical
theory of electricity, except immediate deductions from known formula, that first presented themselves in researches on the the determinafigure of the earth, of which the principal are, tion of the law of the electric density on the surfaces of conducting bodies differing little from a sphere, and on those of ellip-
from 1771, the date of CAVENDISH'S paper, until about 1812, presented to the French Institute two memoirs of singular elegance, relative to the distribution of soids,
when M. PoiSSON
electricity electrified
on the surfaces of conducting spheres, previously and put in presence of each other. It would be quite
there employed. The object of this proposition is to show, that when two similar conducting bodies communicate by means of a long slender canal, and are charged with electricity, the respective quantities of redundant fluid contained in them,
-
n 1 power of their corresponding diameters supposing the electric repulsion to vary inversely as the n power of the distance. This is proved by considering the canal as cylindrical, and filled with incompressible will be proportional to the
fluid of
uniform density
:
:
then the quantities of electricity in the interior of the
two bodies are determined by a very simple geometrical construction, so that the total action exerted on the whole canal by one of them, shall exactly balance that th arising from the other and from some remarks in the 2 7 proposition, it appears the results thus obtained, agree very well with experiments in which real canals are employed, whether they are straight or crooked, provided, as has since been shown by COULOMB, n is equal to two. The author however confesses he is by no means able to demonstrate this, although, as we shall see immediately, it may very ;
easily be
deduced from the propositions contained in this paper. For this purpose, let us conceive an incompressible fluid of uniform density, whose particles do not act on each other, but which are subject to the same actions from all the electricity in their vicinity, as real electric fluid of like density would be
then supposing an infinitely thin canal of this hypothetical fluid, whose per; pendicular sections are all equal and similar, to pass from a point a on the surface of one of the bodies, through a portion of its mass, along the interior of the real canal, and through a part of the other body, so as to reach a point A on its sur-
and then proceed from A to a in a right line, forming thus a closed circuit, it evident from the principles of hydrostatics, and may be proved from our author's d 23 proposition, that the whole of the hypothetical canal will be in equilibrium, face,
is
and as every
particle of the portion contained within the system is necessarily so, the rectilinear portion aA must therefore be in equilibrium. This simple consideration serves to complete CAVENDISH'S demonstration, whatever may be the form or
thickness of the real canal, provided the quantity of electricity in it is very small An analogous application of it will in the bodies.
compared with that contained
render the demonstration of the 22 d proposition complete, when the two coatings of the glass plate communicate with their respective conducting bodies, by fine metallic wires of
any form.
PREFACE.
5
impossible to give any idea of them here
they must be
read.
to
:
It will therefore only
be duly appretiated be remarked, that
they are in fact founded upon the consideration of what have, in this Essay, been termed potential functions, and by means of an equation in variable differences, which may immediately
be obtained from the one given in our tenth article, serving to express the relation between the two potential functions arising
from any spherical surface, the author deduces the values ot these functions belonging to each of the two spheres under consideration, and thence the general expression of the electric density on the surface of either, together with their actions on
any exterior
point.
am
not aware of any material accessions to the theory of electricity, strictly so called, except those before noticed; but I
and magnetic fluids are subject to one common and their theory, considered in a mathematical
since the electric
law of
action,
point of view, consists merely in developing the consequences which flow from this law, modified only by considerations arising
from the peculiar constitution of natural bodies with respect to these two kinds of fluid, it is evident the mathematical theory of the latter, must be very intimately connected with that of the former; nevertheless, because it is here necessary to consider bodies as formed of an immense number of insulated particles, all
acting upon each other mutually,
it
is
easy to conceive that
must, on
this account, present themselves, superior and indeed, until within the last four or five years, no successful difficulties
attempt to overcome them had been published. For this farther extension of the domain of analysis, we are again indebted to
M. PoiSSON, who has already furnished us with three memoirs on magnetism: the first two contain the general equations on which the magnetic state of a body depends, whatever may be its
form, together with their complete solution in case the body is a hollow spherical shell, of uniform thick-
under consideration ness, acted
upon by any
exterior forces,
and
also
when
it
is
a
solid ellipsoid subject to the influence of the earth's action. supposing magnetic changes to require time, although an ex-
By
ceedingly short one, to complete them, that
M. ARAGO'S discovery
relative
to
it
had been suggested
the
magnetic
effects
PREFACE.
6
etc., by rotation, might be M. PoisSON has founded his On hypothesis explained. formulae deduced third memoir, and thence applicable to magthe netism in a state of motion. Whether preceding hypothesis
developed in copper, wood, glass, this
will
serve
M. ARAGO
to explain
or not,
it
the
would
phenomena observed by become me to decide; but it is
singular ill
probably quite adequate to account for those produced by the rapid rotation of iron bodies. have just taken a cursory view of what has hitherto been
We
my knowledge, on subjects connected with the mathematical theory of electricity; and although many of the artifices employed in the works before mentioned are written, to the best of
remarkable for their elegance, only
to particular objects,
it is
easy to see they are adapted
and that some general method, capable
Indeed of being employed in every case, is still wanting. commencement of his in the first memoir (Mem. M. PoiSSON, de V Institute 1811), has incidentally given a method for determining the distribution of electricity on the surface of a spheroid of any form, which would naturally present itself to a person occupied in these researches, being in fact nothing more than the ordinary one noticed in our introductory observations, as requiring the resolution of the equation (a). Instead however of supposing, as we have done, that the point p must be upon the surface, in order that the equation may subsist, M. POISSON availing himself of a general fact, which was then supported by experiment only, has conceived the equation to hold good
wherever this point may be situated, provided it is within the spheroid, but even with this extension the method is liable to the same objection as before. Considering how desirable
it
was
that a
power of universal
agency, like electricity, should, as far as possible, be submitted to calculation, and reflecting on the advantages that arise in the
many difficult problems, from dispensing altogether with a particular examination of each of the forces which actuate
solution of
the various bodies in any system, by confining the attention solely to that peculiar function on whose differentials they all
depend, I was induced to try whether discover
any
it
general relations, existing
would be possible between
to
this function
PREFACE.
and the quantities of
7
electricity in the bodies
advantages LAPLACE had
producing
it.
derived in the third book of the
The Me-
canique Celeste, from the use of a partial differential equation of the second order, there given, were too marked to escape the notice of any one engaged with the present subject, and naturally served to suggest that this equation might be made subservient had in view. Recollecting, after some attempts
to the object I to accomplish
equations,
had
that previous researches on partial differential shown me the necessity of attending to what
it,
have, in this Essay, been denominated the singular values of functions, I found, by combining this consideration with the
method was capable of being apwith plied great advantage to the electrical theory, and was thus, in a short time, enabled to demonstrate the general forpreceding, that the resulting
mulae contained in the preliminary part of the Essay. The as to be remaining part ought regarded principally furnishing particular examples of the use of these general formulas ; their number might with great ease have been increased, but those
which are given,
it is
hoped, will
suffice to point out to
mathe-
maticians, the mode of applying the preliminary results to any case they may wish to investigate. The hypotheses on which the received theory of magnetism is founded, are by no means
which the electrical theory rests; it is however not the less necessary to have the means of submitting them to calculation, for the only way that appears open to us in so certain as the facts on
the investigation of these subjects, which seem as it were desirous to conceal themselves from our view, is to form the most
probable hypotheses we can, to deduce rigorously the consequences which flow from them, and to examine whether such consequences agree numerically with accurate experiments.
The applications of analysis to the physical Sciences, have the double advantage of manifesting the extraordinary powers of this wonderful instrument of thought, and at the same time of serving to increase them truth of this assertion.
M. FOURIER, by
;
numberless are the instances of the
To
select
one
we may remark,
that
his investigations relative to heat, has not only
discovered the general equations on which its motion depends, but has likewise been led to new analytical formulae, by whose
8
PREFACE.
MM. CAUGHT and PoiSSON have been enabled to give the complete theory of the motion of the waves in an indefinitely extended fluid. The same formulae have also put us in posses-
aid
sion of the solutions of
numerous
to
many
be detailed here.
other interesting problems, too must certainly be regarded as
It
a pleasing prospect to analysts, that at a time when astronomy, from the state of perfection to which it has attained, leaves little room for farther applications of their art, the rest of the physical
show themselves
daily more and more willing to submit to it ; and, amongst other things, probably the theory that supposes light to depend on the undulations of a luminiferous sciences should
fluid,
and
to
which the celebrated Dr T.
YOUNG
has given such
furnish a useful subject of research, by affording new opportunities of applying the general theory of the motion of fluids. The number of these opportunities can scarcely
plausibility,
may
be too great, as
it
must be evident
to those
who have examined
the subject, that, although we have long been in possession of the general equations on which this kind of motion depends, we are not yet well acquainted with the various limitations it will to the different
be necessary to introduce, in order to adapt them physical circumstances which may occur.
Should the present Essay tend in any way to facilitate the application of analysis to one of the most interesting of the physical sciences, the author will deem himself amply repaid for
any labour he may have bestowed upon
it
;
and
it is
hoped
the difficulty of the subject will incline mathematicians to read this
work with indulgence, more particularly when they are it was written by a young man, who has been
informed that
obtain the little knowledge he possesses, at such and by such means, as other indispensable avocations which offer but few opportunities of mental improvement,
obliged to intervals
afforded.
INTRODUCTORY OBSERVATIONS.
THE object of this Essay is to submit to Mathematical Analysis the phenomena of the equilibrium of the Electric and Magnetic Fluids, and to lay plicable to perfect
down some general principles equally apand imperfect conductors but, before enter;
ing upon the calculus, it may not be amiss to give a general idea of the method that has enabled us to arrive at results,
remarkable
be very
for their simplicity
difficult if
and generality, which
it
would
not impossible to demonstrate in the ordi-
nary way. It is well
known, that nearly
all
the attractive and repul-
sive forces existing in nature are such, that if we consider any material point p, the effect, in a given direction, of all the point, arising from any system of bodies will be expressed by a partial differential consideration,
forces acting
S under
upon that
of a certain function of the co-ordinates
which serve
to define
the point's position in space. The consideration of this function is of great importance in many inquiries, and probably there are
none in which
its utility is
engage our attention.
more marked than
in those about to
In the sequel we shall often have occasion
speak of this function, and will therefore, for abridgement, call If p be a it the potential function arising from the system S. particle of positive electricity under the influence of forces arising to
from any
electrified
body, the function in question, as
is
well
will be obtained
by dividing the quantity of electricity in each element of the body, by its distance from the particle p> and taking the total sum of these quotients for the whole body,
known,
the quantities of electricity in those elements which are negatively electrified, being regarded as negative.
10
INTRODUCTORY OBSERVATIONS.
by considering the relations existing between the density^ of the electricity in any system, and the potential functions thence arising, that we have been enabled to submit many It is
electrical
phenomena
had hitherto
to calculation /which
resisted
the attempts of analysts and the generality of the consideration here employed, ought necessarily, and does, in fact, introduce a ;
great generality into the
results
obtained from
it.
There
is
one^consideration peculiar to the analysis itself, the nature and utility of which will be best illustrated by the following sketch.
Suppose
it
were required to determine the law of the dis-
A
tribution of the electricity on
a closed conducting surface placed under the influence of any
without thickness, when electrical forces whatever: these forces, for greater simplicity, being reduced to three^X, Y", and Z^& the direction of the rectCIJ.J.VL and CIJ.1 Cll^li.1* IAC*A ^co-ordinates, CV/ AX-LV^i VCIOVJ them. Then \jlJi L/ \SW V**Vfc****W**V*Ji VX/JLfcVUUbU* to increase tending tirgtrhar p (jmfepresenting the density of the electricity on an element dcr of _L
"*!
L
V
JL. JLL
the surface, and r^the distance between dcr and p, any other point of the surface, the equation for determining p which would be employed in the ordinary method, when the problem is re-
duced
to its simplest form, is
known
be --*
to
?.
Ydy + Zdz] the
first
integral relative to dcr extending over the
(a);
whole surface
A, and the second representing the function whose complete differential is Xdx + Ydy + Zdz, x, y and z being the co-ordinates This equation position of
/>,
is
supposed to subsist, whatever
provided
it is
situate
upon A.
may
be the
But we have no
general theory of equations of this description, and whenever we are enabled to resolve one of them, it is because some consideration peculiar to the problem renders, in that particular case, the solution comparatively simple, and must be looked
upon
as the effect of chance, rather than of
any regular and
scientific procedure.
We
will now take a cursory view of the method it is proposed to substitute in the place of the one just mentioned.
INTRODUCTORY OBSERVATIONS. Let us make
B=
I
(Xdx -f Ydy + Zdz) whatever may be
position of the point p,
F=
F and
when p
I
F'=
within the surface A, and the two quantities
11
F',
-
I
is situate
when p
is
the
any where
exterior to it:
although expressed by the same
definite integral, are essentially distinct functions of #, y,
and
z,
the rectangular co-ordinates of p ; these functions, as is well known, having the property of satisfying the partial differential
equations
\
rs=::v'
^2 dx*
If
*
j,,*
T d^ "
Hh
dy*
v
>
A,
^*
'
dz*
V
now we could obtain the values of F and from these equawe should have immediately, by differentiation, the re-
tions,
quired value of p, as will be shown in the sequel. In the first place, let us consider the function F, whose value at the surface is given by the equation (a), since this may be
A
written
the horizontal line over a quantity indicating that it belongs to the surface A. But, as the general integral of the partial differential equation ought to contain two arbitrary functions, some other condition
Now
since
coefficients
is
requisite for the complete determination of F.
F= JI- r
,
it is
can become
within the surface
A
9
evident that none of
infinite
and
it is
when p worthy
is
its differential
situate
any where
of remark, that this
is
precisely the condition required : for, as will be afterwards shown, when it is satisfied we shall have generally
the integral extending over the whole surface, and (p) being a and do-. quantity dependent upon the respective positions of
p
INTRODUCTORY OBSERVATIONS.
12
All the difficulty therefore reduces
V which to the
itself to finding a function the partial differential equation, becomes equal value of at the surface, and is moreover such
satisfies
known
that none of
F
its differential coefficients
shall be infinite
when p
is
within A.
In like manner, in order value at A,
to find F',
by means of the equation
we
shall obtain
V,
its
(a), since this evidently
becomes
a^V'-'B, Moreover
V = I-
it
is
clear, that
can be
infinite
i.e.
~F'=~F
none of the
differential coefficients of
when p
exterior to the surface
is
V
and when^> is at an infinite distance from .4, These two conditions combined with the partial
is
A,
equal to zero.
differential equa-
tion in F', are sufficient in conjunction with its known value at the surface for the complete determination of F', since
A
be proved have
will
hereafter, that
when they
are satisfied
we
V it
shall
the integral, as before, extending over the whole surface A, and a quantity dependent upon the respective position of p (p) being
and da. It only remains therefore to find a function F' which satisfies the partial differential equation, becomes equal to when is
V
p
upon the surface -4, vanishes when p is at an infinite distance from A, and is besides such, that none of its differential coefficients shall be infinite, when the pointy is exterior to A.
whom
the practice of analysis is familiar, will that the problem just mentioned, is far less readily perceive difficult than the direct resolution of the equation (a), and there-
All those to
fore the solution of the question originally proposed has
rendered
much
easier
by what has preceded.
The
sideration relative to the differential coefficients of
been
peculiar con-
F and
F',
by
restricting the generality of the integral of the partial differential equation, so that it can in fact contain only one arbitrary func-
INTRODUCTORY OBSERVATIONS.
13
two which it ought otherwise to have conhas which thus enabled us to effect the simplification tained, and, in question, seems worthy of the attention of analysts, and may be of use in other researches where equations of this nature are tion, in the place of
employed.
We will now give a brief account of what is The
contained in the
seven articles are employed in demonstrating some very general relations existing between the density of the electricity on surfaces and in solids, and the cor-
following Essay.
first
responding potential functions.
These serve as a foundation
to
the more particular applications which follow them. As it would be difficult to give any idea of this part without employing analytical symbols, we shall content ourselves with remarking,
that
it
contains a
generality and
number of singular equations of great
which seem capable of being applied of the electrical theory besides those conmany departments sidered in the following pages. simplicity,
to
In the eighth article we have determined the general values of the densities of the electricity on the inner and outer surfaces of an insulated electrical jar, when, for greater generality, these surfaces are supposed to be connected with separate conductors charged in any way whatever; and have proved, that for the same jar, they depend solely on the difference existing between the two constant quantities, which express the values of the within the respective conductors. Afterwards, from these general values the following consequences have been deduced functions
potential
:
When
an insulated electrical jar we consider only the accumulated on the two surfaces of the glass itself, quantity on the inner surface is precisely equal to that in
electricity
the total
on the outer surface, and of a contrary sign, notwithstanding the great accumulation of electricity on each of them: so that if a communication were established between the two sides of the the sum of the quantities of electricity which would manifest themselves on the two metallic coatings, after the discharge, is exactly equal to that which, before it had taken place, would have been observed to have existed on the surfaces of the coat-
jar,
ings farthest from the glass, the only portions then sensible to the electrometer.
14
INTRODUCTORY OBSERVATIONS.
If an electrical jar communicates by means of a long slender wire with a spherical conductor, and is charged in the ordinary way, the density of the electricity at any point of the interior surface of the jar, is to the density on the conductor itself, as the radius of the spherical conductor to the thickness of the glass in that point.
The
total quantity of electricity contained in the interior of
any number of equal and similar jars, when one of them communicates with the prime conductor and the others are charged by cascade, is precisely equal to that, which one only would receive, if placed in communication with the same conductor, its exterior surface being connected with the
common reservoir. This method when any
of charging batteries, therefore, must not be employed great accumulation of electricity is required.
been shown by M. PoiSSON, in his first Memoir on Magnetism (Mem. de 1'Acad. de Sciences, 1821 et 1822), that It has
when an
placed in the interior of a hollow spherical conducting shell of uniform thickness, it will not be acted upon in the slightest degree by any bodies exterior to the electrified
body
is
however intensely they may be electrified. In the ninth Essay this is proved to be generally true, whatever may be the form or thickness of the conducting shell. In the tenth article there will be found some simple equations, by means of which the density of the electricity induced on a spherical conducting surface, placed under the influence of any electrical forces whatever, is immediately given and thence shell,
article of the present
;
the general value of the potential function for any point either within or without this surface is determined from the arbitrary
value at the surface
itself,
by the
aid of a definite integral.
The
proportion in which the electricity will divide itself between two insulated conducting spheres of different diameters, connected by a very fine wire, is afterwards considered ; and it is
proved, that when the radius of one of them is small compared with the distance between their surfaces, the product of the mean density of the electricity on either sphere, by the radius of that sphere, and again by the shortest distance of its surface from the centre of the other sphere, will be the same for both.
Hence when
their distance is very great, the densities are in the
inverse ratio of the radii of the spheres.
15
INTRODUCTORY OBSERVATIONS.
When
any hollow conducting
shell is
charged with elec-
carried to the exterior surface, tricity, the interior one, as may be on without leaving any portion and fifth articles. In the the fourth from shown immediately
the whole of the fluid
is
it is necessary to leave a small experimental verification of this, it became therefore a problem of some orifice in the shell :
interest to determine the modification
produce.
by
article,
which
this alteration
would
We
have, on this account, terminated the present investigating the law of the distribution of electricity
on a thin spherical conducting shell, having a small circular orifice, and have found that its density is very nearly constant on the exterior surface, except in the immediate vicinity of the
and the density at any point p of the inner surface, is to ; constant the density on the outer one, as the product of the circle into the cube of the radius of the orifice, a of diameter orifice
the product of three times the circumference of that circle into the cube of the distance of p from the centre of the
is to
orifice
;
Hence,
excepting as before those points in its immediate vicinity. if the diameter of the sphere were twelve inches, and
that of the orifice one inch, the density at the point on the inner surface opposite the centre of the orifice, would be less than the
hundred and thirty thousandth part of the constant density on the exterior surface.
In the eleventh
article
some of the
effects
due
to
atmo-
spherical electricity are considered ; the subject is not however insisted upon, as the great variability of the cause which pro-
duces them, and the impossibility of measuring of vagueness to these determinations.
it,
gives a degree
The form of a conducting body being given, it is in general a problem of great difficulty, to determine the law of the distribution of the electric fluid on its surface but it is possible :
of almost every imaginable variety of to bodies such, that the values of the density ; shape, conducting of the electricity on their surfaces may be rigorously assignable
to give
different forms,
the most simple calculations the manner of doing this is explained in the twelfth article, and two examples of its use are
by
:
given. is
In the
last,
the resulting form of the conducting
an oblong spheroid, and the density of the
electricity
body on
its
INTRODUCTORY OBSERVATIONS.
16
surface, here found, agrees with the
one long since deduced from
other methods.
Thus
far perfect conductors
only have been considered.
In
order to give an example of the application of theory to bodies which are not so, we have, in the thirteenth article, supposed the
matter of which they are formed to be endowed with a constant coercive force equal to /5, and analogous to friction in its operation, so that when the resultant of the electric forces acting upon
any one of
their elements is less than
/3,
the electrical state
element shall remain unchanged; but, so soon as it exceed /3, a change shall ensue. Then imagining a to begins solid of revolution to turn continually about its axis, and to be of this
f
acting in parallel "right subject to a constant electrical force electrical state at which the the determine we lines, permanent
The result of the analysis is, that will ultimately arrive. in consequence of the coercive force /3, the solid will receive a new polarity, equal to that which would be induced in it if it
body
were a perfect conductor and acted upon by the constant force to one in the body's equator, making /3, directed in lines parallel the angle 90 + 7, with a plane passing through its axis and being supposed resolved into two parallel to the direction of/ :
/
one in the direction of the body's axis, the other b directed along the intersection of its equator with the plane just
forces,
mentioned, and 7 being determined by the equation
In the latter part of the present article the same problem is considered under a more general point of view, and treated by a different analysis
:
the body's progress from the initial, towards it was the object of the former part to de-
that permanent state
and the great rapidity of
this progress
made
by an example. The phenomena which present themselves during the
rota-
termine
is
exhibited,
evident
tion of iron bodies, subject to the influence of the earth's magnetism, having lately engaged the attention of experimental philosophers, we have been induced to dwell a little on the
solution of the preceding problem, since it may serve in some illustrate what takes place in these cases. Indeed,
measure to
INTRODUCTORY OBSERVATIONS. if
there were
17
any substances in nature whose magnetic powers, and nickel, admit of considerable developement,
like those of iron
which moreover the coercive force was, as we have here supposed it, the same for all their elements, the results of the preceding theory ought scarcely to differ from what would be observed in bodies formed of such substances, provided no one
and
in
of their dimensions was very small, compared with the others. The hypothesis of a constant coercive force was adopted in this article, in
order to simplify the calculations
this is not exactly the case of nature, for steel
has been shown
(I
think by
Mr
probably, however, a bar of the hardest :
Barlow) to have a very
considerable degree of magnetism induced in it by the earth's action, which appears to indicate, that although the coercive
some of
force of
in
which
it
is
its particles is very great, there are others so small as not to be able to resist the feeble
Nevertheless, when iron bodies are turned slowly round their axes, it would seem that our theory ought not to differ greatly from observation and in particular, it is very probable the angle 7 might be rendered sensible to experiaction of the earth.
;
ment, by sufficiently reducing b the component of the force/.
The remaining articles treat of the theory of magnetism. This theory is here founded on an hypothesis relative to the constitution of magnetic bodies, first proposed by COULOMB, and afterwards generally received by philosophers, in which they are considered as formed of an infinite number of conducting elements, separated by intervals absolutely impervious to the magnetic fluid, and by means of the general results contained in the former part of the Essay, we readily obtain the necessary
equations for determining the magnetic state induced in a body of any form, by the action of exterior magnetic forces. These
M. PoiSSON has found by a very des Sciences, 1821 et 1822.) de 1'Acad. (Me*m. If the body in question be a hollow spherical shell of constant thickness, the analysis used by LAPLACE (Mdc. C<51. Liv. 3) equations accord with those
different
is
method.
and the problem capable of a complete
applicable,
whatever
may
forces acting
solution,
be the situation of the centres of the magnetic
upon
we have supposed
After having given the general solution, it. the radius of the shell to become infinite, its
2
INTRODUCTORY OBSERVATIONS.
18
thickness remaining unchanged, and have thence deduced formulae belonging to an indefinitely extended plate of uniform
From these it follows, that when the point p, and thickness. the centres of the magnetic forces are situate on opposite sides of a soft iron plate of great extent, the total action on p will have the same direction as the resultant of all the forces, which would
be exerted on the points p, p p",p", etc. in infinitum if no plate were interposed, and will be equal to this resultant multiplied by a very small constant quantity the points p, p p", p'", &c. ',
:
being plate,
',
on a right line perpendicular to the flat surfaces of the and receding from it so, that the distance between any two all
may be equal to twice the plate's thickness. has just been advanced will be sensibly correct, on the supposition of the distances between the point p and the consecutive points
What
magnetic centres not being very great, compared with the plate's thickness, for, when these distances are exceedingly great, the interposition of the plate will make no sensible alteration in the force with
When
which p is solicited. an elongated body, as a
steel wire for instance, has,
under the influence of powerful magnets, received a greater degree of magnetism than it can retain alone, and is afterwards left to itself, it is
in this state force
we
said to be magnetized to saturation. Now if any one of its conducting elements, the
consider
with which a particle
p
of
magnetism
situate within the
element tends to move, will evidently be precisely equal to its coercive force f, and in equilibrium with it. Supposing there-
be the same for every element, it is clear that the degree of magnetism retained by the wire in a state of saturation, is, on account of its elongated form, exactly the same as would be induced by the action of a constant force, equal to/, fore this force to
directed along lines parallel to its axis, if all the elements were perfect conductors; and consequently, may readily be deter-
mined by the general theory. The number and accuracy of COULOMB'S experiments on cylindric wires magnetized to saturation, rendered
an application of theory
very desirable, in order to compare
it
therefore effected this in the last article,
comparison
is
to this particular case
with experience.
and the
of the most satisfactory kind.
We have
result of the
GENERAL PRELIMINARY RESULTS. THE
function which represents the sum of all the elecacting on a given point divided by their respective distances from this point, has the property of giving, in a very (1.)
tric particles
simple form, the forces by which it is solicited, arising from the electrified mass. shall, in what follows, endeavour
We
whole
between this function, and the density of the electricity in the mass or masses producing it, and apply the relations thus obtained, to the theory of electricity. Firstly, let us consider a body of any form whatever, through to discover
some
relations
which the electricity is distributed according to any given law, and fixed there, and let x, y\ z', be the rectangular co-ordinates of a particle of this body, p the density of the electricity in this particle, so that dx'dydz being the volume of the particle,
p'dxdy'dz shall be the quantity of electricity it contains : moreover, let / be the distance between this particle and a point p exterior to the body, and represent the sum of all the par-
V
their respective distances from co-ordinates are supposed to be cc, y, z, then
ticles of electricity
this point,
shall
whose
divided
by
we have r'
* V(*' - xf + (y' - y)* +
(z'
- z}\
and 1
[p dx'dydz'
-)
r
;
the integral comprehending every particle in the electrified mass
under consideration,
22
GENERAL PRELIMINARY RESULTS.
20
LAPLACE has shown, in his Me*c. Celeste, that the function has the property of satisfying the equation ~ dx*
and as
dz
dtf
z
F
'
equation will be incessantly recurring in what = S F; the it in the abridged form follows, S used in no other sense the whole of symbol being throughout this
we
shall write
this Essay.
In order
by
to prove that
differentiation
= 8 F, we
we immediately
have only
=
obtain
S
to remark, that ,
and conse-
F
substituted for F in the above equaquently each element of it ; hence the whole integral (being considered as
tion satisfies
the
sum
of all these elements) will also satisfy it. This reasonwhen the is within the body, good point p
ing ceases to hold
for then, the coefficients of
into
that
some of the elements which enter
F becoming infinite, it does F satisfies the equation
not therefore necessarily follow
although each of its elements, considered separately, may do so. In order to determine what 8 F becomes for any point within the body, conceive an exceedingly small sphere whose radius is a inclosing the point p at the distance b from its centre, a and b being exceedingly small quantities. Then, the value of
F
may be considered as composed of two parts, one due to the sphere itself, the other due to the whole mass exterior to it but :
the last part evidently becomes equal to zero when substituted in 8 F, we have therefore only to determine the value of for
8
F F for the
small sphere
itself,
which value
is
known
to
be
p being equal to the density within the sphere and consequently to the value of p at p. If now x y z,, be the co-ordinates of the centre of the sphere, we have t ,
t,
GENERAL PRELIMINARY RESULTS.
21
and consequently B
Hence, throughout the interior of the mass
= of which, the equation for any point exterior to the is a particular case, seeing that, here p = 0.
8F
Let now q be any
line terminating in the point p,
without the body, then
\-j) CL\j /
= the
\
body
supposed
force tending to impel a
particle of positive electricity in the direction of q, and tending This is evident, because each of the elements of to increase it.
F substituted
for
F in
)
(-7
,
will give the force arising from
this element in the direction tending to increase
quently,
-T (
)
sum
will give the
of
all
,
and conse-
the forces due to every
element of F, or the total force acting on p in the same direcIn order to show that this will still hold good, although
tion.
F
the point p be within the body ; conceive the value of to be divided into two parts as before, and moreover let p be at the surface of the small sphere or b this small sphere will
= a,
then the force exerted by
be expressed by f
dd>
da being the increment of the radius a, corresponding to the increment dq of q, which force evidently vanishes when a = 0: we need therefore have regard only to the part due to the mass exterior to the sphere, and this is evidently equal to T7-
4-7T
2
V--a*p. But
as the first differentials of this quantity are the same as Fwhen a is made to vanish, it is clear, that whether
those of
the point p be within or without the mass, the force acting it
in the direction of q increasing, is
always given by
upon
GENERAL PRELIMINARY RESULTS.
22
Although in what precedes we have spoken of one body only, the reasoning there employed is general, and will apply equally to a system of any number of bodies whatever, in those cases even, where there is a finite quantity of electricity spread over their surfaces, and it is evident that we shall have for a point p in the interior of any one of these bodies (1).
Moreover, the force tending to increase a line q ending in any point p within or without the bodies, will be likewise given by
(-7)
J
the function
F representing the
sum
of all the electric
particles in the system divided by their respective distances from As this function, which gives in so simple a form the values p.
of electricity, any how particle will recur very frequently in what follows, impelled, ventured to call it the potential function belonging to
of the forces
by which a
p
situated, is
we have
the system, and it will evidently be a function of the co-ordinates of the particle p under consideration.
been long known from experience, that whenever is in a state of equilibrium in any system whatever of perfectly conducting bodies, the whole of the electric fluid It has
(2.)
the electric fluid
will be carried to the surface of those bodies, without the smallest
portion
know
of electricity remaining in their interior: but I do not shown to be a necessary conse-
that this has ever been
quence of the law of electric repulsion, which is found to take This however may be shown to be the case place in nature. imaginable system of conducting bodies, and is an immediate consequence of what has preceded. For let x, y, z, be the rectangular co-ordinates of any particle p in the interior
for every
of one of the bodies: then will
p
is
(-7-
)
\dxj
be the force with which
impelled in the direction of the co-ordinate x, and tending
to increase
forces in
In the same way J
it.
y and
z,
and since the
forces are equal to zero
:
hence
dV and dV -7
dy
7-
dz
will
be the
fluid is in equilibrium all these
GENERAL PRELIMINARY RESULTS.
dV , , = dV -=- ace + -7- ay dx
23
dz
dy
which equation being integrated gives
F=
const.
V
This value of being substituted in the equation preceding number gives ,0
(1)
of the
= 0,
and consequently shows, that the density of the electricity any point in the interior of any body in the system is equal
at to
zero.
The same equation
(1)
will give the value of p the density
of the electricity in the interior of any of the bodies, when there are not perfect conductors, provided we can ascertain the value
F
of the potential function
in their interior.
Before proceeding to
(3.)
make known some
relations
which
between the density of the electric fluid at the surfaces of bodies, and the corresponding values of the potential functions exist
within and without those surfaces, the electric fluid being confined to them alone, we shall in the first place, lay down a general theorem which will afterwards be very useful to us. This theorem may be thus enunciated:
Let
U and F
co-ordinates x, y, infinite at
be two continuous functions of the rectangular z, whose differential co-efficients do not become
any point within a
solid "body of
any form whatever
;
then will
the triple integrals extending over the whole interior of the body, and those relative to d
dw being an
pendicular to the surface, the interior of the body.
infinitely small line per-
and measured from
this surface
towards
GENERAL PRELIMINARY RESULTS.
24
To
prove this
The method
let
.
us consider the triple integral
of integration
by
parts, reduces this to
ay
accents over the quantities indicating, as usual, the values of those quantities at the limits of the integral, which in the present case are on the surface of the body, over whose interior
tlie
the triple integrals are supposed to extend. f
Let us now consider the part
rJ
\dydzV"
TJ" ^
due to the
It is easy to see since dw is every where greater values of x. perpendicular to the surface of the solid, that if d
element of this surface corresponding to dydz, ,
_
dydz
dx
we
,
da
-jdw
,
and hence by substitution
dw In like manner
it is
seen, that in the part ,
dU'
due to the smaller values of x, we shall have -\-
dx j a/(T
-7
aw
,
,
dx
shall
have
25
GENERAL PRELIMINARY RESULTS. and consequently
dx [j v> du> = - t* - IdydzV dv -jw dx } ] '
,
-j
.
dx
Then, since the sum of the elements represented by da-'j together with those represented by dcr", constitute the whole surface of the body, we have by adding these two parts [j J fjr V
\dydz J
\
" dU--
-!
dx
F
T7'
dU =
dx
fj
'\
-j
\dcr-j-
j
dx
J
dw
]
dx
where the integral relative to dcr is supposed to extend over the whole surface, and dx to be the increment of x corresponding to the increment dw. In precisely the same way we have
dw i and
Cj j j
tody
frrndU"
(V
-^
- J7 ,dU'\
F
f,
ds
' ,
dy
~dU
_) = -J^^ F^
;
sum
of all the double integrals in the expression before given will be obtained by adding together the three parts just found; we shall thus have therefore, the
~
dx dw
where
F
+
d
&
dy dw
+}
=-
dz dw}
- r and -7represent the values
[ j
dw
at the surface of the body.
Hence, the integral
dV dU dVdU by using
dVdU
the characteristic 8 in order to abridge the expression,
becomes
i-t-
j dxdydz
VSV.
Since the value of the integral just given remains unchanged
GENERAL PRELIMINARY RESULTS,
26
when we
substitute
clear, that
it
F
-idffU Hence, after
d
~-
I dxdydz
reciprocally,
it
is
mV.
these two expressions of the same quantity,
having changed their signs,
Thus ever
we equate
if
U and
in the place of will also be expressed by
we
shall
have
the theorem appears to be completely established, whatbe the form of the functions and V.
U
may
In our enunciation of the theorem, we have supposed the differentials of and V to be finite within the body under
U
consideration, a condition, the necessity of which does not appear explicitly in the demonstration, but, which is understood in
the method of integration by parts there employed. In order to show more clearly the necessity of this condition, we will now determine the modification which the formula must
U
for example, becomes undergo, when one of the functions, infinite within the body ; and let us suppose it to do so in one
point
p
only
:
moreover, infinitely near this point
sensibly equal to
;
let
U
be
r being the distance between the point p'
and the element dxdydz. Then if we suppose an infinitely small sphere whose radius is a to be described round p, it is clear that our theorem is applicable to the whole of the body exterior to this sphere,
and
since,
BU= & - =
within the sphere,
evident, the triple integrals may still be supposed to extend over the whole body, as the greatest error that this supposition it is
can induce,
is
a quantity of the order a
2 .
Moreover, the part of
r \
dcrU~T~
,
due to the surface of the small sphere
infinitely small quantity of the order
which, since
we have
here
only an
a; there only remains
foV-j due to /fjTT face,
is
this
same
sur-
GENERAL PRELIMINARY RESULTS.
dU dU
%
-J _ ~ -I drz
2
a
4?r F'
becomes
when
27
the radius a
Thus, the equation
supposed to vanish.
is
(2)
becomes
jdxdydzUSV+jda
U^= jdxdydz FS U+jdo- V^~-4ar V
...
(3);
where, as in the former equation, the triple integrals extend over the whole volume of the body, and those relative to cfor, over its
exterior surface
V being the value of Fat the pointy/.
:
In like manner, infinite for
-,
near to the point
p
shall
jdxdydz
such, that
any point p" within the body, and
sensibly equal to
we
F be
the function
if
,
,
U is
infinitely near this point, as it is
evident from
it
is
becomes
moreover, infinitely
what has preceded that
have
m V+jda- U^-lv U"=jdxdydz F8 U+fd
. .
(3');
the integrals being taken as before, and U" representing the becomes infinite. The same value of Z7, at the point p" where
F
process will evidently apply, however great of similar points belonging to the functions
For abridgment, we
shall in
what
may
be the number
U and
F.
follows, call those sin-
gular values of a given function, where its differential coefficients become infinite, and the condition originally imposed upon U and F will be expressed by saying, that neither of them has any
body under
singular values within the solid
(4.)
We
will
now
consideration.
proceed to determine some relations ex-
isting between the density of the electric fluid at the surface of a body, and the potential functions thence arising, within and with-
out this surface.
on an element
For
this, let pdar
da- of the surface,
function for any point
p
within
be the quantity of electricity F, the value of the potential
and
it,
of which the co-ordinates are
GENERAL PRELIMINARY RESULTS.
28 x, y, z.
Then,
p
point
if
v= %,
97,
V be the value of this function
exterior to this surface,
we
shall
P d(T
f
for
any other
have .
f being the co-ordinates of da; and
F/=
pd
f
the integrals relative to the body.
cZcr
extending over the whole surface of
V
It might appear at first view, that to obtain the value of from that of F, we should merely have to change x, y, z into # y', but, this is by no means the case for, the form of the '
:
',
;
potential function changes suddenly, in passing from the space within to that without the surface. Of this, we may give a very
simple example, by supposing the surface to be a sphere whose radius
is
a and centre
at the origin of the co-ordinates
the density p be constant,
which are
With it is
we
shall
;
then, if
have
essentially distinct functions.
F
V
in the general case, and respect to the functions them will satisfy LAPLACE'S equation, and
clear that each of
consequently
= SFand 0=S'F': moreover, neither of them will have singular values; for any and at the point of the spaces to which they respectively belong, surface
itself,
we
shall
have
the horizontal lines over the quantities indicating that they befrom this surface, long to the surface. At an infinite distance
we
shall likewise
have F'
= 0.
GENERAL PRELIMINARY RESULTS.
We
now show,
will
that if any two functions whatever are
taken, satisfying these conditions, to assign one,
29
it
and only one value of
always be in our power which will produce them For this we may remark,
will /o,
for
corresponding potential functions. that the Equation (3) art. 3 being applied to the space within the body, becomes,
=
also S
p
to
-
and S -
which
If now,
,
has but one singular point, viz.
,
V
-
dV
do-
snce
U
by making
=
V belongs, we
:
p
we have
and,
r being the distance between the point
and the element
dcr.
conceive a surface inclosing the
from
;
we
body
at
an in-
it, by applying the formula the surface of the between of the same article to the (2) space surface and this exterior (seeing that here body imaginary
finite distance
-
=U
shall have,
has no singular value)
since the part due to the infinite surface may be neglected, beIn this last equation, it is is there equal to zero.
cause
V
evident that
dw
is
measured from the surface, into the exterior
space, and hence
x
d/~
W/
'
'
+
\^/ \^/
which equation reduces the sum of the two just given
In exactly the same way,
we
shall obtain
;
to
for the point p' exterior to the surface,
GENERAL PRELIMINARY RESULTS.
30
Hence
it
appears, that there exists a value of
_ p =
-
Again,
(fdV\ + (dV'\\
Fand F
7 ,
for the
two potential functions, within
surface.
=
( -w
force
J
with which a particle of positive
electricity p, placed within the surface
impelled
viz.
'
1
~-
which will give and without the
/>,
in the
direction
dw
and directed inwards; and
and
infinitely near
perpendicular to
this
it,
is
surface,
the force with T (~^~ ) expresses
which a similar particle p placed without this surface, on the same normal with p, and also infinitely near it, is impelled outwards in the direction of this normal but the sum of these two forces is equal to double the force that an infinite plane would exert upon p, supposing it uniformly covered with electricity of the same density as at the foot of the normal on which p is and this last force is easily shown to be expressed by 2-Trp, :
;
hence by equating .
4?r
_(?r **n
and consequently there is only one value of p, which can proF and as corresponding potential functions. Although in what precedes, we have considered the surface of one body only, the same arguments apply, how great soever may be their number for the potential functions F and F' would duce
V
;
still
be given by the formulae
the only difference would be, that the integrations must now extend over the surface of all the bodies, and, that the number of
GENERAL PRELIMINARY RESULTS.
31
functions represented by F, would be equal to the number of the bodies, one for each. In this case, if there were given a
value of
F for each
terior space
;
body, together with F' belonging to the exif these functions satisfied to the
and moreover,
above mentioned conditions, it would always be possible to determine the density on the surface of each body, so as to produce these values as potential functions, and there would be but one density, viz. that given by
dw which could do so surface of
(5.)
easy to
dV :
/>,
-j
and
dw
dV -7-7-
belonging to a point on the
any of these bodies.
From what has been before established (art. 3), prove, that when the value of the potential function
it
is
F is
given on any closed surface, there is but one function which can satisfy at the same time the equation
= SF, and the condition, that F shall have no singular values within surface. For the equation (3) art. 3, becomes by sup-
this
posing
&U=0, dw
In this equation,
U is supposed to have only one singular value
within the surface, this point, to
viz. at
the point p' 9 and, infinitely near to
be sensibly equal to -; r being the distance
from p. If now we had a value of U, which, besides satisfying the above written conditions, was equal to zero at the surface itself,
we
should have
U= 0,
and
this equation
would become
GENERAL PRELIMINARY RESULTS.
32
which shows, that
V the value
of
F at
the point
p
is
given,
known. when V To convince ourselves that there does exist such a function as we have supposed Uto be; conceive the surface to be a perits value at the surface is
put in communication with the earth, and a unit
fect conductor
of positive electricity to be concentrated in the point p' then the total potential function arising from p and from the electricity ,
it
will induce
upon the
surface, will be the required value of U.
For, in consequence of the communication established between the conducting surface and the earth, the total potential function at this surface
must be constant, and equal
to that of the earth
to zero (seeing that in this state they form but one itself, conducting body). Taking, therefore, this total potential funci.
tion for
e.
Z7,
we have
_
=U
evidently
those parts infinitely near to^>'.
9
As
no other singular points within the all
the properties assigned to
= 8U
9
1
and
U=-
for
moreover, this function has
surface,
it
evidently possesses
U in the preceding proof.
we have evidently Z7' = 0, for all the space the exterior to surface, the equation (4) art. 4 gives Again, since
of the electricity induced on the surface, (p) is the density the action of a unit of electricity concentrated in the point p.
where
by
Thus, the equation
(5)
of this article becomes
(6).
remarkable on account of its simplicity and it gives the value of the potential for that singularity, seeing any point p, within the surface, when F, its value at the sur-
This equation
is
known, together with (p), the density that a unit of electricity concentrated in p' would induce on this surface, face itself is
if it
conducted electricity perfectly, and were put in communica-
tion with the earth.
Having thus proved, that F' the value of any point p within the surface is
tion F, at
the potential funcgiven, provided
its
GENERAL PRELIMINARY RESULTS. value
V is known it
by
we
will
now show,
that what-
V
V
deduced may be, the general value of the formula just given shall satisfy the equation
ever the value "of
from
at this surface,
33
For, the value of
V at any point p whose
co-ordinates are x, y,
z,
deduced from the assumed value of V, by the above written formula,
is
U being the total potential function within the surface, arising from a unit of electricity concentrated in the point p, and the electricity
since
T
is
induced on the surface
itself
by
evidently independent of x, y,
its
action.
Then,
we immediately
z,
deduce
U
Now the general value of will depend upon the position of the point p producing it, and upon that of any other point p whose co-ordinates are x, y ', z, to which it is referred, and will consequently be a function of the six quantities, x, y^
But we may conceive
U
to
z,
x, y
be divided into two parts, one
',
z.
=-
r
being the distance pp ) arising from the electricity in p, the other, due to the electricity induced on the surface by the action (r
U
U
has no singular Then since oi'p, and which we shall call values within the surface, we may deduce its general value from that at the surface, by a formula similar to the one just given. .
t
l
Thus
where U'
is
the total potential function, which would be pro-
duced by a unit of
electricity inj/,
pendent of the co-ordinates x y, 9
and z,
therefore,
(-T-
is
inde-
)
of p, to which 8 refers.
GENERAL PRELIMINARY RESULTS.
34
Hence
We have before supposed
and as 8 - = 0, we immediately obtain
Again, since
we have
at the surface itself
r being the distance between
and the element
p
do-,
we hence
deduce
this substituted in the general value of 8Z7, before given, there The result just obarises 8Z7, 8?7. 0, and consequently
=
=
tained being general, and applicable surface, gives
to
any point p" within the
immediately
w and we have by substituting in the equation determining 8F,
= 8F. In a preceding part of
this article,
we have
obtained the
equation
which combined with
=8
and therefore the density is
(-7-
(p)
J
,
gives
induced on any element
evidently a function of the co-ordinates
a?,
y, z,
of
<7
^>,
which is
also
GENERAL PRELIMINARY RESULTS.
35
such a function as will satisfy the equation = 8 (p) it is moreover evident, that (p) can never become infinite when p is within :
the surface. It
now remains
For surface
this, ;
then
it,
prove that the formula
V = F,
any point within the surface and whatever may be the assumed value of V.
shall always give infinitely near
to
for
suppose the point it is
p
to
approach infinitely near the
clear that the value of (p)
,
the density of the
induced by p, will be insensible, except for those parts infinitely near to p, and in these parts it is easy to see, that the value of (p) will be independent of the form of the surelectricity
and depend only on the distance p, dcr. But, we shall afterwards show (art. 10), that when this surface is a sphere of any
face,
radius whatever, the value of
(p) is
a being the shortest distance between p and the surface, and/ This expression will give an (p) decreases, in passing from
representing the distance p, da-. idea of the rapidity with which
the infinitely small portion of the surface in the immediate vicinity of p, to any other part situate at a finite distance from it, and when substituted in the above written value of F, gives,
by supposing a
to vanish,
F=F. It is also evident, that the function F, determined by the above written formula, will have no singular values within the surface
under consideration.
What was before surface, may likewise
proved, for the space within any closed
be shown to hold good, for that exterior to a number of closed surfaces, of any forms whatever, provided shall be equal to zero at an we introduce the condition, that
V
infinite distance
from these surfaces.
For, conceive a surface at
32
GENERAL PRELIMINARY RESULTS.
36 an
from those under consideration
infinite distance
we have
before said,
may
;
then,
what
be applied to the whole space within
the infinite surface and exterior to the others
;
consequently
where the sign of integration must extend over all the surfaces, (seeing that the part due to the infinite surface is destroyed by is there equal to zero), and dw must evithe condition, that
V
dently be measured from the surfaces, into the exterior space to
V
now belongs. The form of the equation
which
(6)
remains also unaltered, and (6');
the sign of integration extending over all the surfaces, and (p) being the density of the electricity which would be induced on
each of the bodies, in presence of each other, supposing they all communicated with the earth by means of infinitely thin conducting wires.*
Let now A be any closed surface, conducting electricity and p a point within it, in which a given quantity of electricity Q is concentrated, and suppose this to induce an then will F, the value of the potential electrical state in -A function arising from the surface only, at any other pointy', also within it, be such a function of the co-ordinates p and -p', that we may change the co-ordinates of p, into those of p', and (6.)
perfectly,
;
Or, in other words, the reciprocally, without altering its value. value of the potential function at p' due to the surface alone, when the inducing electricity Q is concentrated in p, is equal to 9
that which
would have place
at
j?,
if
the same electricity
Q
were
concentrated in p.
Fof, in consequence of the equilibrium at the surface, we have evidently, in the first case, when the inducing electricity is concentrated inj?,
* In connexion with the subject of this article, see a paper by Professor Thomson, Cambridge and Dublin Mathematical Journal, Vol. vi. p. 109.
GENERAL PRELIMINARY RESULTS.
37
r being the distance between p and da-' an element of the surface ft a constant quantity dependent upon the quantity of
A, and
electricity originally placed
Now
on A.
F at p
the value of
is
by what has been shown
(art. 5) ; (//) being, as in that article, of the electricity which would be induced on the elethe density ment da by a unit of electricity in p ', if the surface were put
A
This equation gives
in communication with the earth.
since S
F=
S
But we know
x, y, z, of p.
similar
way
the symbol 8 referring to the co-ordinates
=0;
= S' F;
that
to the co-ordinates x', ?/,
0',
where
8'
refers in
a
of j?' only.
Hence we
F has
no singular
have simultaneously
where
it
must be remarked, that the function
and are both situate within values, provided the points the surface A. This being the case the first equation evidently
p
gives
p
(art. 5)
V=-j(p)daV;
V
being what
carried to da,
x, y, z, and
F
would become,
p remaining f,
function of x, y,
rj,
z,
f,
f
,
if
the inducing point
Where
fixed.
F
independent of
f
x-,
y
,
z
were
a function of
is
the co-ordinates of da, whereas ?/,
p
;
(p)
is
a
hence by the
second equation
-j(p)da$V, which could not hold generally whatever might be the of p, unless we had
situation
= SF; where we must be cautious, not
to
confound the present value of
GENERAL PRELIMINARY RESULTS.
38 F,
with that employed at the beginning of this
the equation
= 8 F,
which
last,
article in
having performed
proving
its office,
will
be no longer employed.
The
= 8' V
equation
V being what F becomes
gives in the same
the point
by bringing
element dv of the surface A.
way
This substituted
p
to
any other
for V, in the ex-
pression before given, there arises
a'V':
V=+fj(p)(P")dcd in
which double
integral, the signs of integration, relative to each
da and dv, must extend over
of the independent elements
whole
the
surface.
p
tion at
we
represent by F', the value of the potential funcarising from the surface A, when the electricity Q is
If now,
concentrated in
p, we
shall evidently
have
where the order of integrations alone
is
changed, the limits re-
j~
V
: being what F, would become, by first bringing the electrical point p to the surface, and afterward the
maining unaltered
p to which
point
t
V
t
This being done,
belongs.
it is
clear that ~V'
T~
and F represent but one and the same quantity, seeing that each of them serves to express the value of the potential function, /
at
any point of the
when
the electricity
surface
is
trified point, situate in
hence
we have
A,
any other point of the same
evidently
F as
was
arising from the surface itself, it by the action of an elec-
induced upon
asserted at the
F
~ */
commencement
of this article.
surface,
and
GENERAL PRELIMINARY RESULTS. from
39
that our preceding arguments will be equally applicable to the space exterior to the surfaces of any number of conducting bodies, provided we introduce the conIt is evident
art. 5,
that the potential function F, belonging to this space, shall remove to an inshall be equal to zero, when either or finite distance from these bodies, which condition will evidently dition,
p
p
be
satisfied,
state.
provided
Supposing
all
the bodies are originally in a natural be the case, we see that the
this therefore to
potential function belonging to
any point p of the
exterior space,
arising from the electricity induced on the surfaces of any number of conducting bodies, by an electrified point in p, is equal to
that which would have place at p,
removed
if
the electrified point were
to p'.
What
has been just advanced, being perfectly independent
number and magnitude of the conducting bodies, may be applied to the case of an infinite number of particles, in each of which the fluid may move freely, but which are so of the
cannot pass from one to another. This is always supposed to take place in the theory of magand the present article will be found of great use to
constituted that
what
is
it
netism, us when in the sequel (7.)
tions,
to treat of that theory.
These things being established with respect to electrified the general theory of the relations between the denthe electric fluid and the corresponding potential func-
surfaces sity of
we come
;
when
the electricity
is
disseminated through the interior
of solid bodies as well as over their surfaces, will very readily
flow from
For
what has been proved
(art. 1).
V
represent the value of the potential function at a point p', within a solid body of any form, arising from the whole of the electric fluid contained in it, and p be the density
of the
this let
electricity in
its
interior;
p being a function of the if p be the density at
three rectangular co-ordinates x, y, z then the surface of the body, we shall have :
T7 ,_ [dx
dy dz p
~r~^~
+
[clcrp
)~>
r being the distance between the point p whose co-ordinates are
GENEEAL PRELIMINARY RESULTS.
40
and that whose co-ordinates are longs, also r the distance between p and x, y\ z
,
surface of the
If
now
it is
F
clear
body be what
:
do-,
V being evidently a function
V becomes by changing x
from
art. 1,
which p bean element of the
#, y, z, to
that p will be given
',
of
x',
y
,
z
.
z into x, y, z, #',
by
Substituting for p the value which results from this equation, in that immediately preceding we obtain ',
fr,
V
- - --
[dxdydzSV
=
I
J
:
which, by means of the equation
[pdar \
J
.the
7
4WT
r
-
[pd
,
I
J
r
(3) art. 3,
j
becomes
1 [, v\~r [ - (dV -- [do- 1=747rJ\da-v dw J r \ \
\
1
horizontal lines over the quantities indicating that they be-
long to the surface
itself.
F, to be the value of the potential function in the exterior to the body, which, by art. 5, will depend on space at the surface only; and the equation (2) the value of = art. 3, applied to this exterior space, will give, since
Suppose
F
8F
y
and 8 r
= 0,
where dw' is measured from the surface into the exterior space to which F, belongs, as dw is, into the interior space. Conse= and therefore dw dw, quently
41
GENERAL PRELIMINARY RESULTS. Hence the equation determining p becomes, by
its
substituting for
value just given,
[p^_^fd^(idV\ r
]
r
"47rJ
|U*/
(dV\\
W/J'
an equation which could not subsist generally, unless
dV Thus the whole difficulty is reduced to finding the value the potential function exterior to the body.
V
t
of
Although we have considered only one body, it is clear that the same theory is applicable to any number of bodies, and that the values of p and p will be given by precisely the same formulae, however great that number may be: V being the exterior potential function common-to all the bodies. t
In case the bodies under consideration are ductors,
we have
^eeii (art. 1), that the
will be carried to their surfaces,
all perfect
whole of the
and therefore there
con-
electricity is
here no
place for the application of the
theory contained in this article ; but as there are probably no perfectly conducting bodies in
becomes indispensably necessary, if we would electrical the phenomena in all their generality. investigate nature, this theory
Having in this, and the preceding articles, laid down the most general principles of the electrical theory, we shall in what follows apply these principles to more special cases; and the necessity of confining this Essay within a moderate extent, will compel us to limit ourselves to a brief examination of the more interesting
phenomena.
APPLICATION OF THE PRECEDING RESULTS TO THE THEORY OF ELECTRICITY.
THE
first application we shall make of the foregoing will be to the theory of the Leyden phial. For this, principles, we will call the inner surface of the phial A, and suppose it to be
(8.)
any form whatever, plane or curved, then, B being its outer surface, and 6 the thickness of the glass measured along a normal to A 6 will be a very small quantity, which, for greater of
;
generality, we will suppose to vary in to another. one point of the surface
A
any way, If
in passing
from
now
the inner coating of the phial be put in communication with a conductor C, charged with any quantity of electricity, and the outer one be also made to communicate with another conducting body (7', containing any other quantity of electricity, it is evident, in consequence of the communications here established, that the total potential function,
arising
from the whole system, will
be constant throughout the interior of the inner metallic coating,
and
of the
quantity
body
(7.
We
shall
here represent this constant
by #.
Moreover, the same potential function within the substance of the outer coating, and in the interior of the conductor (?', will
be equal
to another constant quantity ff.
Then
designating by F, the value of this function, for the whole of the space exterior to the conducting bodies of the system,
APPLICATION OF THE PRECEDING RESULTS, &C.
43
and consequently for that within the substance of the glass we shall have (art. 4)
itself;
and
F=/3 One
F = #'.
horizontal line over
any quantity indicating that it belongs A, and two showing that it belongs to the
to the inner surface
outer one B.
At any drawn, and
point of the surface
A, suppose a normal
be the axes of
w
to it to
be
w", being two other rectangular axes, which are necessarily in the plane tanat this point; gent to may be considered as a function let this
then
:
w',
V
A
of w, w and w", and we shall have by TAYLOR'S theorem, since and w" = at the axis of w along which 6 is measured, w' =
dw
dw z
1
1
2
where, on account of the smallness of 0, the series converges and their very rapidly. By writing in the above, for
V
we
values just given,
#-
V
obtain
e d*v 0* = dv 3=-7 + 3=TT-^
dw
I
1.2
dw'
In the same way, if w be a normal to B, directed towards -4, and 6 be the thickness of the glass measured along this normal, we shall have t
ft
-f,.
dw
1
dw2 1-2
But, if we neglect quantities of the order 0, compared with those retained, the following equation will evidently hold good,
dwn
dwn n being any whole troduced because
Now by
article
w
positive
and
w
number, the factor
dw
l)
being in-
are measured in opposite directions.
4
= dV ^
w (
= .
and
47T/>
= dV = dw
APPLICATION OF THE PRECEDING RESULTS
44
p and p being the densities of the
A
B
and
electric fluid - at the surfaces
Permitting ourselves, in what follows,
respectively.
to neglect quantities of the order 0* compared with those reand hence by tained, it is clear that we may write 6 for /?
substitution
where
V
and p are quantities of the order
the order $ or unity.
be determined,
is
the value of
/3
being
d*V -=- for
any point on the
to
sur-
A.
Throughout the substance tion
$ and
The only thing which now remains dw*
face
7=;
V will
of the glass, the potential funcB F, and therefore at a point
satisfy the equation
=
on the surface of A, where of necessity w, w' and w" are each equal to zero, we have
dw2
dw*
dw" z
mark over w, w and w" Then since w = 0,
the horizontal omitted.
and as
V
is
constant and equal to
ft
being, for simplicity,
at the surface
A, there
hence arises
w*
R
being the radius of curvature at the surface A, in the plane
w }. f
(w,
dV
Substituting these values in the
diately preceding,
we
get
~ w'*
E
dw
expression imme-
TO THE THEORY OF ELECTRICITY. In precisely the same way we obtain, radius of curvature in the plane (w, w"},
by writing
45
R
for the
both rays being accounted positive on the side where w, i. e. w, These values substituted in = F, there results is negative.
dw*
and thus the sum of the two
for the required value
of -r- *->
equations into which
enters, yields
and the
it
difference of the
same equations gives
-# = 29r(p-p)0; therefore the required values of the densities p
and p are
2 which values are correct to quantities of the order 6 p, or, which is the same thing, to quantities of the order 0; these having
been neglected in the
unworthy
latter part of the
preceding analysis, as
of notice.
do- is an element of the surface A, the corwill be of B, cut off by normals to element responding
Suppose
da-
jl
last
+
6 (-ft
A
+
gHj-
element will be
,
and
~pdcr
9
therefore the quantity of fluid
l
+6
-^
+-
;
on
this
substituting for p ita
APPLICATION OF THE PEECEDING RESULTS
46
value before found,
we
&*pj
3 /
= ~/
l~~0~D + 7pf> an ^-
neglecting
obtain
- pda, the same quantity as on the element da- of the first surface. therefore, we conceive any portion of the surface A, bounded
If,
by
a closed curve, and a corresponding portion of the surface B, which would be cut off by a normal to A, passing completely round this curve the sum of the two quantities of electric fluid, on these corresponding portions, will be equal to zero and con;
;
sequently, in an electrical jar any how charged, the total quantity of electricity in the jar may be found, by calculating the quanfarthest tity, on the two exterior surfaces of the metallic coatings
from the
glass, as the portions of electricity,
on the two surfaces This result
adjacent to the glass, exactly neutralise each other.
will appear singular, when we consider the immense quantity of fluid collected on these last surfaces, and moreover, it would not
be
difficult to verify it
by experiment.
As
a particular example of the use of this general theory suppose a spherical conductor whose radius a, to communicate :
electrical jar, by means of a long slender the outside wire, being in communication with the common reservoir ; and let the whole be charged then representing the density of the electricity on the surface of the conductor,
with the inside of an
:
P
which
will be very nearly constant, the value of the potential function within the sphere, and, in consequence of the communication established, at the inner coating also, will be 4-TraP very nearly, since we may, without sensible error, neglect the
A
action of the wire
and jar /3
and the equations
(8),
itself in calculating it.
= 4?raP by
and f
Hence
= 0,
neglecting quantities of the order
6,
give
We
thus obtain,
by
the most simple calculation, the values of
TO THE THEORY OF ELECTRICITY.
47
A
and B, the densities, at any point on either of the surfaces that on the when conductor is next the glass, known. spherical
The theory
of the condenser, electrophorous, &c. depends has been proved in this article ; but these are details
upon what which the
into
there
is,
limits of this
however, one
Essay will not permit
result, relative to
me
to enter
;
charging a number of
jars ~by cascade, that appears worthy of notice, and which flows so readily from the equations (8), that I cannot refrain from introducing it here.
Conceive any number of equal and similar insulated Leyden uniform thickness, so disposed, that the exterior coatphials, of ing of the first may communicate with the interior one of the second the exterior one of the second, with the interior one of the third; and so on throughout the whole series, to the ex;
which we will suppose in communication with the earth. Then, if the interior of the first phial be made to communicate with the prime conductor of an electrical
terior surface of the last,
all the phials will receive a certain of operating is called charging by cascade. Permitting ourselves to neglect the small quantities of free fluid on the exterior surfaces of the metallic coatings, and other quan-
machine, in a state of action,
charge, and
this
mode
the same order, we may readily determine the electrical state of each phial in the series: for thus, the equations (8) tities of
become
_
p=
Designating now, by an index at the foot of any letter, the number of the phial to which it belongs, so that, p^ may belong to the
first,
p 2 to the second phial, and so on ; we shall have, by whole number to be n, since 6 is the same for
supposing their every one,
1
"-""fe" &c.
APPLICATION OF THE PRECEDING RESULTS
48
ffn-ffn ~_ P*~ 47T0
Now
/3
= _ fin ~ A ~
Pn
'
47T0
represents the value of the total potential function,
within the prime conductor and interior coating of the
first
phial,
and in consequence of the communications established in this system, we have in regular succession, beginning with the prime conductor, and ending with the exterior surface of the last phial,
which communicates with the
earth,
= A+^; o=fr + fr;
&c. ...o
= ^_ + 1
=p +p
But the first system of equations gives whole number s may be, and the second hibited
two
is
expressed
by
= p a-1 + p
8
;
/3 n
8
8
.
,
whatever
line of that just ex-
hence by comparing these
last equations,
which shows that every phial of the system is equally charged. Moreover, if we sum up vertically, each of the columns of the first
system, there will arise in virtue of the second /Q
I^'
_
_ Q
We
therefore see, that the total charge of all the phials is precisely the same, as that which one only would receive, if
placed in communication with the same conductor, provided its Hence this exterior coating were connected with the earth.
mode
of charging, although it may save time, will never produce a greater accumulation of fluid than would take place if one phial only were employed. (9.)
Conceive
now
a hollow shell of perfectly conducting
matter, of any form and thickness whatever, to be acted upon
by any
electrified bodies, situate
without
it
;
and suppose them
to
TO THE THEORY OF ELECTRICITY. induce an electrical state in the shell
be such, that the placed any where within state
For
V
let
total action
;
40
then will this induced
on an
electrified particle, will be it, absolutely null. the of value the total potential function, represent
any point p within the shell, then we surface, which is a closed one, at
have
shall
at its inner
being the constant quantity, which expresses the value of the potential function, within the substance of the shell, where the electricity is, by the supposition, in equilibrium, in virtue of the
(3
combined with that arising from
actions of the exterior bodies,
the electricity induced in the shell itself. Moreover, V evidently = B F, and has no singular value within satisfies the equation the closed surface to which it belongs : it follows therefore, from
Art
5,
that
its
general value
is
Y*& and as the
forces acting
upon p are given by the t
differentials of
F, these forces are evidently all equal to zero. If, on the contrary, the electrified bodies are all within the shell,
and
earth,
it
exterior surface is put in communication with the equally easy to prove, that there will not be the
its is
on any electrified point exterior to it ; but, the action of the electricity induced on its inner surface, by the electrified bodies within it, will exactly balance the direct action slightest action
of the bodies themselves.
Or more generally
:
Suppose we have a hollow, and perfectly conducting shell, bounded by any two closed surfaces, and a number of electrical bodies are placed, some within and some without it, at will then, if the inner surface and interior bodies be called the interior ;
system system
;
;
also, the outer surface all
and exterior bodies the exterior
phenomena of the repulsions, and densities,
the electrical
interior system,
same would take place if there were no exterior system, and the inner surface were a perfect conductor, put in communication with the earth and all those of the exterior system will be the same, as if the interior one did not exist, and the outer surface were a
relative to attractions,
will be the
as
;
perfect conductor, containing a quantity of electricity, equal to
4
APPLICATION OF THE PRECEDING RESULTS
50
the whole of that originally contained in the shell all the interior bodies.
This
so direct a consequence of what has been 5, that a formal demonstration would
is
and
in
shown
in
itself,
and
articles 4
be quite it is easy to see, the as which could difference only superfluous, where to the interior between the case exist, relative system, there
is
an exterior system, and where there
is
not one, would
be in the addition of a constant quantity, to the total potential function within the exterior surface, which constant quantity must necessarily disappear in the differentials of this function,
and consequently, in the values of the attractions, densities, which all depend on these differentials
repulsions, and In the alone.
exterior system there is not even this difference, but the total potential function exterior to the inner surface is precisely the same, whether we suppose the interior system to exist or not.
The
consideration of the electrical phenomena, which arise from spheres variously arranged, is rather interesting, on ac(10.)
count of the ease with which
may
be put to the
test of
all
the results obtained from theory the complete solution ; but,
experiment
of the simple case of two spheres only, previously electrified, and put in presence of each other, requires the aid of a profound analysis, and has been most ably treated by M. PoiSSON (Mem.
de
1'Institut. 1811).
Our
object, in the present article, is
merely
two examples of determinations, relative to the distribution of electricity on spheres, which may be expressed by
to give one or
very simple formulae.
Suppose a spherical surface whose radius is a, to be covered with electric matter, and let its variable density be represented
by p
;
then
function
if,
as in the Me*c. Celeste,
V, belonging to a point
p
we expand
the potential
within the sphere, in the
form
r being the distance between p and the centre of the sphere, and {0 (1 \ \ etc. functions of the two other polar co-ordinates of p,
U
U
it is clear,
by what has been shown
in the admirable
work
just
TO THE THEORY OF ELECTRICITY.
51
mentioned, that the potential function V, arising from the same spherical surface, and belonging to a point p, exterior to this distance r from its centre,
surface, at the
and on the radius r
produced, will be
r = Z7 If,
therefore,
tions
and
(0)
%+
U
(l)
we make V=
-|r
(r),
and
+ etc.
(2)
V
^ (/),
the two func-
will satisfy the equation r
* But
~ + Z7
(art. 4)
dv dV and the equation between
dv <
and
-v/r,
in its first form, gives,
by
differentiation,
Making now
^>'
of
r
=a
and ty' being the characteristics of the differential and o/r, according to Lagrange's notation.
co-efficients
In the same
is
there arises
way
the equation in
its
second form yields
These substituted successively^ in the equation by which p we have the following,
determined,
dr
a (9).
dr
a
therefore, the value of the potential function be known, either for the space within the surface, or for that without it, the If,
42
APPLICATION OF THE PRECEDING RESULTS
52
value of the density p will be immediately given, of these equations.
From what
has preceded,
we may
by one
readily determine
electric fluid will distribute itself, in a
or other
how
the
conducting sphere whose
when
acted upon by any bodies situate without it In this case, the electrical state of these bodies being given. we have immediately the value of the potential function arising radius
is a,
;
from them. Let this value, for any point p within the sphere, be represented by A A being a function of the radius r, and two other polar co-ordinates. Then the whole of the electricity will be carried to the surface (art. 1), and if Fbe the potential ;
function arising from this electrified surface, for the same point p, we shall have, in virtue of the equilibrium within the sphere,
V+A=ft ft
being a constant quantity.
or
V=ft~A;
V being
substituted
that the
quantities
This value of
in the first of the equations (9), there results
47T/0
A $ dA--= - n2 -7 +-: a ar a
the horizontal lines indicating, as before,
under them belong to the surface itself. In case the sphere communicates with the earth, ft is evidently equal to zero, and p is completely determined by the above but if the sphere is insulated, and contains any quantity :
Q of electricity, the value of ft may be ascertained as follows : Let be the value of the potential function without the surface,
V
corresponding to the value
V=
ft
A
within
it;
then,
by what
precedes
A' being determined from
and
r',
A by
the following equations
being the radius corresponding to the point
:
p', exterior
TO THE THEORY OP ELECTRICITY. to the sphere, to
V=
evidently
-7
When
which A' belongs. .
Therefore
r being made infinite. of p becomes known.
r
53
is finite,
we have
by equating
Having thus the value
of $, the value
To
give an example of the application of the second equation us suppose a spherical conducting surface, whose radius is a, in communication with the earth, to be acted upon by any bodies situate within it, and B' to be the value of the potential
in p
let
The function arising from them, for a point p exterior to it. total potential function, arising from the interior bodies and will evidently be equal to zero at this surface, and Hence consequently (art. 5), at any point exterior to it. of due surface. second to the Thus the j?'=0; being
surface
itself,
V+
V
the equations
(9)
becomes
4?rp
=2
+ ,
-j-r
dr
.
a
We
are therefore able, by means of this very simple equation, to determine the density of the electricity induced on the surface in question.
Suppose now
all the interior
bodies to reduce themselves to
a single point P, in which a unit of electricity is concentrated, the potential function arising from
and /to be the distance Pp
P will be
-j, ,
:
and hence
j l
TV
=/' r being, as before, the distance between p and the centre of Let now b represent the distance OP, and 6 the the shell.
angle POp' equation
,
then will
we deduce
f =b 2
2
- 2Jr
.
cos
successively,
r'-i cos e
+ r\
From which
54
APPLICATION OF THE PRECEDING RESULTS
and
2
Making order to
=a
and in the value of B' before given, in obtain those which belong to the surface, there results r
f
in this,
dB'
*
dr'
+
B' a
2a _=
2
+ 2ab cosfl + f _ V-a* af af .
This substituted in the general equation written above, there arses
If
P
is
supposed to approach infinitely near to the surface, = a a ; a being an infinitely small quantity, this
so that b
would become cc
A
In the same way, by the aid of the equation between and the density of the electric fluid, induced on the surface of a is exterior when the electrified point sphere whose radius is
p,
P
,
to
it,
is
found to be
supposing the sphere to communicate, by means of an infinitely with the earth, at so great a distance, that we might
fine wire,
neglect the influence of the electricity induced upon it by the action of P. If the distance of from the surface be equal
P
to
an
infinitely small quantity
in the foregoing,
o P
=
we
a,
shall
have in
this case, as
-
a
27T.
From what has preceded, we may readily deduce the general value of F, belonging to any point P, within the sphere, when V its value at the surface is known. For (p), the density induced upon an element
do- of the surface,
concentrated in P, has just been
shown
y-a
2
3 ;
47m/
to
by a be
unit of electricity
TO THE THEORY OF ELECTRICITY.
f
being the distance P, (6), art. 5,
equation
dcr.
55
This substituted in the general
gives
In the same way we shall have, when the point the sphere,
P is exterior to
-* fa }
,
f
}
The use of these two equations will appear almost immediately, when we come to determine the distribution of the electric fluid,
on a thin spherical
shell, perforated
with a small circular
orifice.
The results just given may be readily obtained by means of LAPLACE'S much admired analysis (Mec. Ce'l. Liv. 3, Ch. n.), and indeed, our general equations (9), flow very easily from the Want of room compels equation (2) art. 10 of that Chapter. me to omit these confirmations of our analysis, and this I do the more freely, as the manner of deducing them must immediately occur to any one
who has
read this part of the Me'-
canique Celeste. Conceive now, two spheres S and /S", whose radii are a and a, to communicate with each other by means of an infinitely it is fine wire required to determine the ratio of the quantities :
of electric fluid on these spheres, when in a state of equilibrium supposing the distance of their centres to be represented by b.
The tricity
;
value of the potential function, arising from the elecsurface of S, at a point ^?, placed in its
on. the
centre, is
da-
being an element of the surface of the sphere, p the density and Q the total quantity- on the
of the fluid on this element, If now sphere. function for the
we
represent
by
JF",
the value of the potential
same point p, arising from
by adding together both parts,
S we f
,
shall have,
APPLICATION OF THE PRECEDING RESULTS
56
the value of the total potential function belonging to p, the In like manner, the value of this function at p centre of S. t
the centre of S', will be
F being
the part arising from 8, and
electricity
on
But
S'.
Q
the total quantity of
in consequence of the equilibrium of the
system, the total potential function throughout a constant quantity. Hence
its
whole
interior
is
F
Although it is difficult to assign the rigorous values of and F'; yet when the distance between the surfaces of the two spheres is considerable, compared with the radius of one of them, and F' will be Very nearly the same, it is easy to see that as if the electricity on each of the spheres producing them was
F
concentrated in their respective centres, and therefore
we have
very nearly
F=%o
and ^'=-f-\ o
These substituted in the above, there
Thus
arises
Q to Q' is given by a very simple equation, be the form of the connecting wire, provided it be
the ratio of
whatever
may
a very fine one. If we wished to put this result of calculation to the test of and P' for the experiment, it would be more simple to write
P
mean
densities of the fluid
on the spheres, or those which would being connected as above, they were
be observed when, after separated to such a distance, as not sensibly.
Then
since
Q = 4?ra P 2
we have by
and Q'
substitution, etc.
P_
a
(b
- a)
F~a'(1>-a'
to
influence
each other
TO THE THEORY OF ELECTRICITY.
We
therefore see, that
when
the distance
57
between the centres
Z>
of the spheres is very great, the mean densities will be inversely as the radii and these last remaining unchanged, the density ;
on the smaller sphere will decrease, and that on the larger increase in a very simple way, by making them approach each other.
Lastly, let us endeavour to determine the law of the distrifluid, when in equilibrium on a very thin
bution of the electric spherical shell, in
which there
is
a small circular
orifice.
Then,
we
neglect quantities of the order of the thickness of the shell, compared with its radius, we may consider it as an infinitely
if
S
thin spherical surface, of which the greater segment is a perfect conductor, and the smaller one s constitutes the circular In virtue of the equilibrium, the value of the potential orifice.
on the conducting segment, will be equal to a constant quantity, as F, and if there were no orifice, the corresponding value of the density would be
function,
a being the radius of the spherical surface.
Moreover on
this
supposition, the value of the potential function for any point P,
within the surface, would be F.
Let
therefore,
-
W
4?ra
+p
re-
present the general value of the density, at any point on the surface of either segment of the sphere, and V, that of the cor-
F+
The value of the responding potential function for the point P. potential function for any point on the surface of the sphere will be
F+ V,
whole of
which equated segment
to F, its value
on
$, gives for the
this
0=F. Thus
the equation (10) of this article becomes
the integral extending over the surface of the smaller segment which, without sensible error, may be considered as a
s only,
plane.
APPLICATION OF THE PRECEDING RESULTS
58
But, since
it
evident that p
is
to the potential function V,
segment
s,
dw
it
is
the density corresponding have for any point on the
is
shall
treated as a plane,
P as
we
easy to
see,
~_-leZF dw
27T
'
from what has been before shown
(art. 4)
;
being perpendicular to the surface, and directed towards the
centre of the sphere ; the horizontal line always serving to in"When the point dicate quantities belonging to the surface.
P P
very near the plane s, and z is a perpendicular from upon s, z will be a very small quantity, of which the square is
Thus
and higher powers may be neglected.
b
=a
and by
z,
substitution
the integral extending over the surface of the small plane Now being, as before, the distance P, do:
s,
and
f
= ~~
dw at the surface of
s,
and
^
dz
= ~~X
^dV'_~ldV'_--l
f>
d_
we suppose
z
at the
=
[zdo-
2^~dw"-~*jr^z~~te?dz) provided
nence
f
d*
1
"47T
2
=
[dv
d#]J Now
end of the calculus.
the
p
density zero,
--
H
/o,
upon the
and therefore we have
surface of the orifice for the
whole of
5,
equal to
is
this surface
F Hence by
substitution
F
'
l
the integral extending over the whole of the plane
an element, and z being supposed equal the operations have been effected. da- is
s,
(12). } ' (
of
which
to zero, after all
TO THE THEORY OF ELECTRICITY. It
now only remains For
to
59
V from
determine the value of
this
now
equation. represent the linear radius of s, and y, the distance between its centre C and the foot of the perpendicular z then if we conceive an infinitely thin oblate this, let /3
:
spheroid, of uniform density, of which the circular plane s constitutes the equator, the value of the potential function at the point P, arising from this spheroid, will be
T)
The
being the distance do; 0, and k a constant quantity.
attraction exerted
pendicular
z,
by
~~ and by
will be
to the attractions of
this spheroid, in the direction of the per-
the
,
known
formulae relative
homogeneous spheroids, we have
.
M representing the mass of the spheroid, and 6 being determined by
the equations
tan 6
=-
.
a
Supposing now z very small, since it is to vanish at the end of the calculus, and y < /3, in order that the point may fall within the limits of s, we shall have by neglecting quantities of z the order z compared with those retained
P
and consequently
V)
This expression, being differentiated again relative d*
,
fdo-
SMir
to Zj gives
APPLICATION OF THE PRECEDING RESULTS
60
But the mass
M
is
given by
M=Jcj Hence by
substitution
d2 dz
which expression if
is
t/
= 0. Comparing rigorously exact when z of the (12) present article, we see
with the equation
this result
that
z
V=
~k
*J (13*
if) ,
the constant quantity
In
determined, so as to satisfy (12).
fact,
Ik
may be always
we have only
to
make 77*
27
7T
K
=
T7T
J77T
.
1.
a
Having thus the value
Jb
7
K
.
.
air
of F, the general value of
V
is
known,
since yjr
^M
\s
i
i4/is
Cu
-w-^-
~~
(s
CL
I
(JjU
f
7
~TT"
P
The value of the potential function, for any point within the shell, being V, and that in the interior of the conducting matter of the shell being constant, in virtue of the equilibrium, the value p of the density, at any point on the inner surface of
F+
the shell, will be given immediately art. 4.
by the general formula
(4)
Thus ,
P
-1 dV dV = +F -J-=A4?r db =T~ ^T T-T4-Tr aw iara 1
.
( s
tan
-
0)
which equation, the point P is supposed to be upon the element d
TO THE THEORY OF ELECTRICITY,
we
have R* = y*
shall
8Z order
-
and by neglecting quantities of the
compared with those retained, we have successively
=
a -.8,
Thus
-f z*,
61
'
the value of
and
|,
tan0~0 =
i0'
becomes
In the same way,
is easy to show from the equation (11) the value of the density on an element da-" of the exterior surface of the shell, corresponding to the
of this article, that
element
da-'
it
/>",
of the interior surface, will be
which, on account of the smallness of p for every part of the surface, except very near the orifice s, is sensibly constant and equal to
W ,
therefore
p"
37T.E
3
'
which equation shows how very small the density within the shell
is,
even when the
orifice is considerable.
The
determination of the electrical phenomena, which result from long metallic wires, insulated and suspended in the (11.)
atmosphere, depends upon the most simple calculations. As an and B, connected by a example, let us conceive two spheres then slender wire; representing the conducting pdxdydz long
A
quantity of electricity in an element dxdydz of the exterior in the vicinity of the space, (whether it results from the ground
wire having become slightly electrical^ or from a mist, or even _a passing cloud,) and r being the distance of this element from
A's centre;
also
r'
its
distance from
Z?'s,
the value of the
APPLICATION OF THE PRECEDING RESULTS
92
potential function at -4's centre, arising from the space, will be
whole exterior
*
pdxdydz
and the value of the same function
at j&'s centre will be
[pdxdydz
P
J
'
the integrals extending over all the space exterior to the conducting system under consideration.
Q
If now,
be the
total quantity of electricity
on A's surface,
and
Q' that on J5's, their radii being a and a' ; it is clear, the value of the potential function at -4's centre, arising from the system itself, will be
seeing that) we may neglect the part due to the wire, on account of its fineness^ and that due to the other sphere, on account of In a similar way, the value of the same function its distance. at j5's centre will be found to be
a
But
(art* 1)
the value of the total potential function must be
constant throughout the whole interior of the conducting system, and therefore its value at the two centres must be equal hence ;
Q -a
f pdxdydz
/
j_
J
in the present case, is exceedingly small, the />, in this equation may not only be considerable, contained integrals but very great, since they are of the second dimension relative
Although
to space.
other,
may
The spheres, when at a great distance from each therefore become highly electrical, according to the
observations of experimental philosophers, and the charge they will receive in any proposed case may readily be calculated ; the value of p being supposed given. When one of the spheres,
TO THE THEORY OF ELECTRICITY.
B
63
connected with the ground, Q' will be equal and consequently Q immediately given. If, on the contrary, the whole system were insulated and retained its natural quantity of electricity, we should have, neglecting that on the for instance, is
to zero,
wire,
and hence
Q
and Q' would be known.
were required to determine the electrical state of the when in communication with a wire, of which one extremity is elevated into the atmosphere, and terminates in If
it
sphere A,
a fine point p,
we
and consequently,
Hence
should only have to make the radius of B> vanish in the expression before given.
Q',
in this case
Q_
f
pdxdydz
f pdxdydz
^
"^J r being the distance between p and the element dxdydz. Since the object of the present article is merely to indicate the cause of
some phenomena of atmospherical electricity, it to a greater length, more particularly
extend
difficulty of determining
correctly the
it
is
useless to
as the
electrical
state
extreme of the
any given time, precludes the possibility of putatmosphere ting this part of the theory to the test of accurate experiment. at
(12.)
Supposing the form of a conducting body
to
be given,
in general impossible to assign, rigorously, the law of the density of the electric fluid on its surface in a state of equilibrium,
it is
when not acted upon by any exterior bodies, and, at present, there has not even been found any convenient mode of approximation It is, however, extremely easy to applicable to this problem. such forms to conducting bodies, that this law shall be give by the most simple means. The following method, depending upon art. 4 and 5, seems to give to these forms the greatest degree of generality of which they are sus-
rigorously assignable
ceptible, as,
by a
tentative process,
approximated indefinitely.
any form whatever might be
APPLICATION OF THE PRECEDING RESULTS
64
Take any continuous ordinates x,
y',
= 8F',
ferential equation
an
function
z, of a point
p
',
F', of the rectangular cosatisfies the partial dif-
which
and vanishes when p'
is
removed
to
from the origin of the co-ordinates. Choose a constant quantity &, such that may be the
infinite distance
V
V=b
may have no sinequation of a closed surface A, and that is as so this exterior surface: then if to long p gular values, a conducting body, whose outer surface density of the electric fluid in equilibrium upon
we form
represented
is it,
A, the will be
by dw'
47T
'
potential function due to this fluid, for to the body, will be exterior
and the
1
any point
p
,
AF'; h being a constant quantity dependent upon the total quantity communicated to the body. This is evident of electricity from what has been proved in the articles cited. Let R represent the distance between p, and any point ,
within
A
tricity
upon
;
then the potential function arising from the elec-
be expressed by
will
it
-r>,
when
R
is
infinite.
Hence the condition
Q = hV which will serve
(R being
to determine A,
infinite),
when Q
is
given.
In the application of this general method, we may assume for F', either some analytical expression containing the coordinates of p,
and
to vanish
which
when p
is
known
is
to satisfy the equation
removed
the origin of the co-ordinates
;
to
an
= 8F',
infinite distance
as, for instance,
from
some of those
given by LAPLACE (M^c. Celeste, Liv. 3, Ch. 2), or, the value a potential function, which would arise from a quantity of elecwithin a finite space, at a point p' tricity anyhow distributed without that space ditions to
which F'
;
since this last will always satisfy the conis subject.
TO THE THEORY OF ELECTRICITY. It
may
In the
65
proper to give an example of each of these cases. place, let us take the general expression given by
"be
first
LAPLACE,
then,
by confining ourselves
V
value of
two
to the
first
terms, the assumed
will be
r being the distance of p from the origin of the co-ordinates, (0} (l) &c. functions of the two other polar co-ordinates and
U
6 and axes,
U
,
tzr.
may
,
This expression by changing the direction of the always be reduced to the form IT V
i
_
^ cos ^
^a I '
r
a and k being two constant
Then
r
2
quantities,
which we
will suppose
be a very small positive quantity, the form positive. of the surface given by the equation b, will differ but little if b
V
from a sphere, whose radius
is
-,-
:
by gradually
the difference becomes greater, until b
form assigned by
Making
therefore
=
^
;
increasing
and afterwards, the
F=&, becomes improper for our purpose. b = in order to have a surface differing as
p
,
much from
a sphere, as the assumed value of surface becomes of the equation
V admits,
the
A
-r
From which we
r ,_2a
now
If cos
_ a*
obtain r
If
b,
=
represents the angle formed ,
-dr
by dr and dw, we have
Q V 2 sin y 2
2^2 cos?
APPLICATION OF THE PRECEDING RESULTS
66
and as the electricity is in equilibrium upon A, the force with which a particle p, infinitely near to it, would be repelled, must be directed along consequently
dw
:
its effect
ing to increase
it,
but the value of this force
in the direction of the radius
dV' -jr cos
will be
by
equally represented
dV is
-7
,
This
(f>.
-j-j r,
and
,
and tend-
last quantity is
and therefore
--dV' = - dV' -j-r cos 6 -j
dw
dr
;
the horizontal lines over quantities, indicating, as before, that
they belong to the surface
duced from this equation,
itself.
The value
of
f
-jr J
,
de-
is /i
._
_
dw ~~cos$
dr
cos<
this substituted in the general value of p, before given, there arises
47T
dw'
2
2?rr cos
the quantity of electricity communicated to the surface, the condition
Supposing
Q
is
= hV
-^
(where
R is infinite),
before given, becomes, since r may here be substituted for E, seeing that it is measured from a point within the surface,
Q^aTi
7"
We
Q
:
~7"
"2^'
have thus the rigorous value of p for the surface A whose kz / Q\ 1 + ^2 cos when the quantity Q of elecequation is r = ) Q, 2/ (
\
TO THE THEORY OF ELECTRICITY. tricity
upon
it
is
known, and by substituting
67
for r
and h
their
values just given, there results
p=
2
4
COS
47T/C
6
+ J2
1
(
COS 2
\
Moreover the value of the potential function whose polar co-ordinates are r, 0, and CT, is '
=
7+
GA
-
From which we may immediately deduce any pointy
exterior to
for the point p'
the forces acting on
A.
In tracing the surface A, 6 is supposed to extend from 6 = = TT, and -GT, from ur = to OT = 2?r it is therefore evident,
to
6
by
constructing the curve whose equation
:
that the parts about P, where 6
= TT,
form towards a cone whose apex electricity at
P is
null, in the
is
approximate continually in
P, and as the density of the example before us, we may make is
when any body whatever has a part of surface in the form of a cone, directed inwards ; the density of the electricity in equilibrium upon it, will be null at its apex, this general inference
:
its
precisely the reverse of what would take place, rected outwards, for then, the density at the apex
if it
were di-
would become
infinite*. * Since this was written, I have obtained formulae serving to express, geneof a cone, rally, the law of the distribution of the electric fluid near the apex which forms part of a conducting surface of revolution having the same axis.
From these formulae it results that, when the apex of the cone is directed inwards, the density of the electric fluid at any point p, near to it, is proportional to rn-1 ; r being the distance Op, and the exponent n very nearly such as would satisfy the simple equation (4^ + 2) j8=3?r If 2/3 exceeds TT, this summit
where
2/3 is the angle at the summit of the cone. directed outwards, and when the excess is not very considerable, n will be given as above but 2j8 still increasing, until it becomes 27T-27 ; the angle 27 at the summit of the cone, which is now directed :
is
:
outwards, being very small, n will be given by in log ducting body
is
a sphere whose radius
is 6,
on which
= i,
and
in case the con-
P represents the mean
density
52
APPLICATION OF THE PRECEDING RESULTS
68
r
a second example, we will assume for P ', the value of the potential function arising from the action of a line uniformly covered with electricity. Let 2a be the length of the line, y the
As
perpendicular falling from any point p' upon it, x the distance of the foot of this perpendicular from the middle of the line, and x that of the element dx from the same point then taking :
the element dx ', as the measure of the quantity of electricity will be contains, the assumed value of
it
V
V- (
-WS -aa
dx> 2
~J\% +(*-*Tr
a to x = + a. Making the integral being taken from x this equal to a constant quantity log 5, we shall have, for the equation of the surface A,
which by reduction becomes
= y (i - IJ + x\ 4Z> (1 - IY -
We
thus see that this surface
revolution of an ellipsis about transverse axis being
a
&
">*
(1
+
2 )
-
a spheroid produced by the greatest diameter ; the semi-
is its
1+b -f* T=l-P>
and semi-conjugate
differentiating the general value of 9 substituting for y its value at the surface
By
A
_
afj/'
_ O 1-5 '1 + 6
V
,
we
just given,
and
obtain
- 2a/3a;
of the electric fluid, p, the value of the density near the apex 0, will be determined
by the formula
a being the length
~
of the cone.
T
'
n-l
TO THE THEORY OF ELECTRICITY.
Now
writing
ds
1
cos
for the
<j>
_
7/1
b
I
1
2x *Jb \/
dy
(f)
angle formed by dx and
(
2)
2
bj
'
On
the surface
of p
A
<
example, the general value
therefore, in this
_-hdV'_ ~ ~ 4?r
ah$
dw
and the potential function
27T7
for
Making now x and y both
exterior to
any pointy',
infinite, in
order that
A,
is
p may
be at
infinite distance, there results
and thus the condition determining electricity upon the surface, is, since to
Hence, as in
ellipsis.
is
p
an
aV*)
dV' dx
1
cos
_ V(/3
j
ds being an element of the generating the preceding example, we shall have
dw
dw\ we have 4
+ &y
\l
69
vo^+y
These
E
in
* ne quantity of be supposed equal
Q,
may
1
)*
Q
known
A,
i,
tali i.e.
ri
= Q
.
results of our analysis agree with what has been long concerning the law of the distribution of electric fluid on
the surface of a spheroid,
when
in a state of equilibrium.
In what has preceded, we have confined ourselves to (13.) will now give an the consideration of perfect conductors. of our of the general method, to a body that example application
We
APPLICATION OF THE PRECEDING RESULTS
70
supposed to conduct electricity imperfectly, and which will, moreover, be interesting, as it serves to illustrate the magnetic phenomena, produced by the rotation of bodies under the inis
fluence of the earth's magnetism.
If
any
solid
body whatever of revolution, turn about its axis, determine what will take place, when the matter
required to of this solid is not perfectly conducting, supposing it under the influence of a constant electrical force, acting parallel to any a given right line fixed in space, the body being originally in
it is
natural state.
Let
/3
designate the coercive force of the body, which
we
will suppose analogous to friction in its operation, so that as long as the total force acting upon any particle within the body is
less
when
it
than
begins
In the
we
/?, its
to
electrical state shall
exceed
/3,
remain unchanged, but
a change shall ensue.
suppose the constant electrical force, which by &, to act in a direction parallel to a line
first place,
will designate
passing through the centre of the body, and perpendicular to and let us consider this line as the axis its axis of revolution ;
of x, that of revolution being the axis of z, and y the other rectangular co-ordinate of a point p, within the body and fixed in space. Thus, if for the same point electricity of the
V be the value of the total potential function
at any instant of time, arising from the the exterior force, and body jo,
bx
+V
since will be the part due to the body itself at the same instant bx is that due to the constant force J, acting in the direction :
of x, and tending to increase z
= r cos O
the angle
t
x
If
it.
= r sin
cos
now we make r,
y
= r sin 6 sin
-or
;
being supposed to increase in the direction of the body's revolution, the part due to the body itself becomes r
br sin 6 cos
Were we
-or
+
V.
V
suppose the value of the potential function given at any instant, we might find its value at the next instant, to
TO THE THEORY OF ELECTRICITY.
71
conceiving, that whilst the body moves forward through the infinitely small angle da, the electricity within it shall remain fixed, and then be permitted to move, until it is in equilibrium
by
with the coercive force.
Now the body
the value of the potential function at p, arising from itself, after having moved through the angle da* (the
electricity VT into CT
being fixed), will evidently be obtained by changing dco in the
br sin 6 cos
expression just given, and tzr
+ V'+ br sin 6 sin w dw
is
therefore
dV dco.
7
din-
adding
now
bodies,
and restoring
the part
bx x, y,
dV_
=
br sin 6 cos
&c.
we
dV
dx
due
to the exterior
dV
y dx 4 x dy y
r,
have, since
'
dy)
for the value of the total potential function at the
end of the
We
next instant, the electricity being still supposed fixed. have now only to determine what this will become, by allowing the electricity to move forward until the total forces acting on points within the body, which may now exceed the coercive force by an infinitely small quantity, are again reduced to an equilibrium with it. If this were done, we should, wT hen the initial state of the
body was
given, be able to determine, successively,
its
But since it is state for every one of the following instants. that the body, by reevident from the nature of the problem, volving, will quickly arrive at a permanent state, in which the value of Fwill afterwards remain unchanged and be independent of its initial value, we will here confine ourselves to the It is easy to see, by determination of this permanent state. total potential funcfrom the new forces the arising considering that in this case the has been value whose given, tion, just electricity will
be in motion over the whole interior of the body,
and consequently
APPLICATION OF THE PRECEDING RESULTS
72
which equation expresses that the
total force to
move any
par-
ticle p, within the body, just equal to ft, the coercive force. if we can assume any value for V, satisfying the above, is
Now
and such, that belonging
to the
it
shall
new
itself
reproduce
after
the electricity
total potential function (Art. 7), is
with the coercive
to find its equilibrium
allowed
force, it is evident this
will be the required value, since the rest of the electricity is exactly in equilibrium with the exterior force Z>, and may therefore
To be
be here neglected.
conceive two
Y
new axes
able to do this the
f
in
X', and making the angle 7 with them ,
potential function, before given,
V+dco.
/,
+ bx
(by cos 7
which, by assuming
V= fty',
more
advance of the old ones
easily, Jf,
then the value of the
;
Y,
new
becomes
- x dV\ + y',dV -p-p J ,
sin
7
,
and determining 7 by the equa-
tion
=b reduces
sin
7
ft,
itself to
y Considering
belonging
now
(ft
+
b cosydco).
the symmetrical distribution of the electricity with regard to the plane
to this potential function,
whose equation
is
= y',
it
will be evident that, after the elec-
tricity has found its equilibrium, the value of V at this plane must be equal to zero : a condition which, combined with the
partial differential equation before given, will serve to determine, at the next instant, and this value of completely, the value of
V
V will
be
V=fty.
We
thus see that the assumed value of
the end of the following instant, and belonging to the permanent state.
is
V
reproduces
itself at
therefore the one required
If the body had been a perfect conductor, the value of V would evidently have been equal to zero, seeing that it was sup-
posed originally in a natural state that just found is therefore due to the rotation combined with the coercive force, and we :
TO THE THEORY OF ELECTRICITY. thus see that their effect of
y
positive,
is to
polarise the
making the angle
TT 4-
body
73
in the direction
7 with the
direction of
and the degree of polarity will be the same as would be produced by a force equal to /3, acting in this direction on a perfectly conducting body of the same the constant force J;
dimensions.
We
have hitherto supposed the constant force
to act in a
direction parallel to the equatorial plane of the body, but whatever may be its direction, we may conceive it decomposed into
two
one equal to b as before, and parallel to this plane, the ; other perpendicular to it, which last will evidently produce no effect on the value of F, as this is due to the coercive force, and
would if
the
be equal to zero under the influence of the new body conducted electricity perfectly. still
force,
Knowing the value of the potential function at the surface of the body, due to the rotation, its value for all the exterior space may be considered as determined (Art. 5), and if the body be a solid sphere,
may
easily be expressed analytically
;
for it is
F just
evident (Art. 7), given, that even in to the surface confined the present case all the electricity will be of the solid; and it has been shown (Art. 10), that when the
from the value of
value of the potential function for the point surface,
whose radius
is a, is
p
within a spherical
represented by
the value of the same function for a point p, situate without this sphere, on the prolongation of r, and at the distance r from its centre,
will be
But we have seen the point
;/,
that the value of
F due
to the rotation, for
is
F=/3/=/3rcos<9';
&
being the angle formed
the ray r and the axis of
by
the corresponding value for the pointy/ will therefore be ,
V
_ /3a
3
cos(9' To
y ';
APPLICATION OF THE PRECEDING RESULTS
74
And
hence, by differentiation, we immediately obtain the value of the forces acting on any particle situate without the sphere,
rotation but, if we would determine the from the sphere, we must, to the value of the potential function just found, add that part which would be
which
arise
from
its
;
total forces arising
produced by the action of the constant force upon this sphere, when it is supposed to conduct electricity perfectly, which will
be given in precisely the same way as the former. In fact,/ designating the constant force, and 6" the angle formed by r and a line parallel to the direction of/, the potential function arising it, for the point JP, will be
from
- r/cos 0", and consequently the part arising from the by its action, must be
electricity,
induced
+fr cos 0", The corseeing that their sum ought to be equal to zero. to the the exterior value for point p sphere, is responding ',
therefore 3
fa cos 0"
this
added
to the
value of
before found, will give the value from the /?', arising
V,
of the total potential function for the point
sphere
itself.
It will
be seen when we come to
treat of the theory of
magnetism, that the results of his theory, in general, agree very nearly with those which would arise from supposing the magnetic fluid at liberty to
move from one
part of a magnetized body whose magnetic powers admit of the iron and nickel for example
to another; at least, for bodies
considerable developement, as ; errors of the latter supposition being of the order 1 g only ; the of the nature on a constant body, quantity dependant g being which in those just mentioned, differs very little from unity. It is therefore evident that
when
a solid of revolution, formed
of iron, is caused to revolve slowly round its axis, and placed the act of under the influence of the earth's magnetic force
/
revolving, combined with the coercive force
fi
of the body, will
TO THE THEORY OF ELECTRICITY.
75
new
polarity, whose direction and quantity will be very nearly the same as those before determined. /having been supposed resolved into two forces, one equal to b in the
produce a
Now
plane of the body's equator, and another perpendicular to this plane if {3 be very small compared with 5, the angle 7 will ;
new
be very small, and the direction of the
very nearly at right angles to the direction of
polarity will be ,
a result which
has been confirmed by many experiments but by our analysis we moreover see that when b is sufficiently reduced, the angle :
7 may be rendered
and the direction of the new polarity + 7 7 being deter-
sensible,
will then form with that of b the angle JTT
;
mined by the equation sin
7=^
This would be very easily put
.
to the test of
experiment by em-
ploying a solid sphere of iron.
The
values of the forces induced
rotation of the body, which would be observed in the space exterior to it, may be obtained by differentiating that of before given, and will be found to agree with the observations of Mr BARLOW (Phil. Tran.
by the
V
1825), on the supposition of
As the experimental mena developed by the
j3
being very small.
investigation of the magnetic phenorotation of bodies, has lately engaged
the attention of several distinguished philosophers, it may not be amiss to consider the subject in a more general way, as we shall thus not only confirm the preceding analysis, but be able
show with what rapidity the body approaches that permanent state, which it has been the object of the preceding part of this to
article to determine.
Let us now, therefore, consider a body A fixed in space, under the influence of electric forces which vary according to any given law then we might propose to determine the elec;
of the body, after a certain interval of time, from the knowledge of its initial state supposing a constant coercive trical
state
;
To
its most general would to be those parts between form, necessary distinguish of the body where the fluid was at rest, from the forces acting
force to it
exist within
it.
resolve this in
APPLICATION OF THE PRECEDING RESULTS
76
there being less than the coercive force, and those where it would be in motion ; moreover these parts would vary at every
and the problem therefore become very
instant,
we however
intricate
were
:
to suppose the initial state so chosen, that the total
move any particle p within A, arising from its electric and exterior actions, was then just equal to the coercive
force to state
force
/3
;
that the alteration in the exterior forces should
also,
always be such, that if the electric fluid remained at rest during the next instant, this total force should no where be less than the problem would become more easy, and still possess a For in this case, when the fluid is great degree of generality. moveable, the whole force tending to move any particle p within
/3;
will, at every instant, be exactly equal to the coercive force. If therefore #, y, z represent the co-ordinates of p, and the value of the total potential function at any instant of time t,
A,
V
arising from the electric state of the shall have the equation
body and
exterior forces,
we
.
+
,
-7-
\dy ) whose general
integral
may
f.I-T-J
\dz
............
)
be thus constructed
f
()
:
V arbitrarily over any surface whatever and suppose three rectangular co-ordinates w, w w", whose origin is at a point P on S: the axis of w being a normal to S, and those of w', w", in its plane tangent. Take the value
of
S, plane or curved, ',
Then
the values of -7, and
the value of
which
-7-7,
dw
dw
is
dV -7 dw
are
will be determined
known
at the point P,
and
by the equation
merely a transformation of the above.
Take now another point
dV dV
these axes are -7-
dw
,
-7, dw
,
and
P,,
whose co-ordinates
dV -7-77
dw
,
referred to
i r and draw a ri^ht line ,
,
LT
TO THE THEORY OF ELECTRICITY.
P
77
V
then will the value of through the points P, lf on be L, expressed by p,
at
any point
F.+0X; X being
the distance Pp, measured along the line L, considered as increasing in the direction PP, and F the given value of ,
,
V at P. by this and is
For
it is
very easy to see that the value of
V furnished
construction, satisfies the partial differential equation (a), its general integral ; moreover the system of lines
L", &c. belonging to the points P, P', P", &c. on S, are evidently those along which the electric fluid tends to move, and
L,
L',
move during the following instant. Let now V+DV represent what V becomes at the end the time t + dt substituting this for V in (a) we obtain will
of
;
Q
Then,
if
we
= dV dDV
designate
dDV dV dDV
dV.
'
'
dx' dx
dy
by D'
dz
dy
V, the
'
dz
augmentation of the potential
change which takes place in the arising from exterior forces during the element of time dt, the
function,
DV-D'V be the increment of the potential function, due to the corresponding alterations Dp and Dp in the densities of the electric
will
fluid at the surface of
mined from
DV
A
D'Fby
and within Art.
7.
partial differential equations, the
it,
But,
which
by
the
may
be deter-
known theory
most general value of
of
DV satis-
fy ing (5), will be constant along every one of the lines L, L', L", &c., and may vary arbitrarily in passing from one of them to
another
:
as
it
is also
along these lines the electric fluid moves
during the instant dt, it is clear the total quantity of fluid in any infinitely thin needle, formed by them, and terminating in the opposite surfaces of A, will undergo no alteration during this Hence therefore instant.
(c);
dv being an element of the volume of the needle, and da, d& the two elements of A's surface by which it is terminated. This l ,
APPLICATION OF THE PRECEDING RESULTS
78
combined with the equation (b), will completely determine the value of DV, and we shall thus have the value of the
condition,
V+ D V,
potential function value F, at the time
t,
is
at the instant of time
t
+ dt, when
its
known.
As an
application of this general solution suppose the body a solid of revolution, whose axis is that of the co-ordinate z, and let the two other axes X, Y, situate in its equator, be If now the exterior electric forces are such that fixed in space.
A
;
is
may be reduced to two, one equal to c, acting parallel to other equal to b, directed parallel to a line in the plane the z, the variable angle (f> with X; the value of the (xy), making they
from the exterior forces, will be potential function arising
xb cos <j)yb sin
zc
where b and
and
c are constant quantities,
/3
(x cos
+ y sin w)
r
varies with the
When
be constantly increasing. time to t, suppose the value of V to be so as to
V
;
the time
is
equal
:
then the system of lines L, L', L" will make the angle w with the plane (xz), and be perpendicular to another plane whose equation
is
= x cos
r
+ y sin
r.
4- D<j>, the augIf during the instant of time dt, becomes mentation of the potential function due to the elementary change <
in the exterior forces, will be
jy 7= (x sin moreover the equation
= cos
(j>
;
becomes
(b)
dDV -sr
- y cos <) bD<j>
.
-y
dx
sin vf
F-
.
7
,,
[01
dy
and therefore the general value of
D F= DF (y cos
dDV
. ,
D V is
r
x sin
;);
7
jD^ being the characteristic of an infinitely small arbitrary funcV tion. But, it has been before remarked that the value of
D
TO THE THEORY OF ELECTRICITY. will be completely determined,
and the condition
satisfying the equation
by
(b)
Let us then assume
(c).
= hD(j>
x
sin
being a quantity independent of x, y, z, and see determine h so as to satisfy the condition
(c).
DF(y cos ST 7*.
79
x sin
OT
z)
;
(y cos
or
sible to
-or)
;
be poson
if it
Now
this supposition
DV
D'
V
The
x sin
hD
= Z> {y value of
+
(h cos ?r
Dp
(# sin
CT)
x
& cos <)
(h sin
or
<
?/
+
cos
)
fo&
b cos $)}.
corresponding to this potential function
(Art. 7)
Dp =
is
0,
and on account of the parallelism of the lines L, L', &c. to each The condition (c) thus other, and to -4's equator da = da-^ becomes .
(c):
Dp and Dp
being the elementary densities on A*s surface at ends of any of the lines L, L', &c. corresponding to opposite But it is easy to see from the potential function }
DVD'V.
the form
i
of this function, that these elementary densities at ends of any line perpendicular to a plane whose equaopposite tion is = y (h cos r + b cos (/>) x (h sin r -f- b sin <j>) ,
i
are (c)
equal and of contrary signs, and therefore the condition by making this plane coincide with that
will be satisfied
perpendicular to L, L',
marked,
that
is
&c.,
whose equation, as before
re-
is
= x cos w + y sin CT
the condition
;
will be satisfied, if
(c)
h be determined by
the equation
h cos
VF
+b
sin
cos
(j)
__
h sin
-or
+ & sin $
cos
-cr
which by reduction becomes
= h + J cos
(
w),
r
APPLICATION OF TH.E PRECEDING RESULTS
80
and consequently
V+D V= = $B
(x cos OT
ft
+
OT -jcos
+ y sin
sin OT cos
-f /:ty
= fixcos
- -5 cos
jw
1
(<
When D(j>,
therefore
is
cos
V
Ztyh
TO-)
((/>
,
-~
JOT
cos
augmented by the
remains unaltered
;
.
-cr)
(<
Ztyj-
infinitely small angle -5
cos
w)
(<
Z>(/>,
the preceding reasoning and the general re-
consequently applicable to every instant,
lation
between
and
<
OT
expressed by
= DOT + -5 a
-cr
receives the corresponding increment
-cr
and the form of is
cos
Ztyl
-BT)
sin
H- /%/
-
sin is
sin OT)
D<j>[
-or)
(<
-^
x
hD<j) (y cos CT
-f
tsr)
common
- OT) D$
(<
:
which by integration gives
differential equation,
77
H being
cos
sn
an arbitrary constant, and
7, as in the
former part of
this article, the smallest root of
Let
OT O
and
<
be the
remained
fixed,
V = ft t
whole
are x, y,
z,
TO-
and
if
the electric fluid
;
then the
would be
(x cos
arid the
values of
next instant,
initial
total potential function at the
vTfi
+ y sin vr + (x sin
force to
)
move a
particle p,
7 cos
)
bd
whose co-ordinates
TO THE THEORY OF ELECTRICITY.
81
which, in order that our solution may be applicable, must not OTO must be be less than ft, and consequently the angle between and TT when this is the case, OT is immediately determined from by what has preceded. In fact, by finding the :
<
value of II from the initial values
?= i?r + iy + i*r n ?-. ^-
e
10, we
and
<
,
f.
have, in the latter part of this article, considered the and the line X', parallel to the direction of 5, :
X'
the relative motion of state of the
pending on
made,
as in the former,
if,
body
to
X, F, Z, evidently de-
motion only, will consequently remain In order to determine it on the supposition
let
X' be the
axis of x, one of the co-ordinates of p, z, the other
referred to the rectangular axes JT', F', Z, also y, two ; the direction X' F', being that in which if
way, so that
X may remain unaltered, the electric
referred to the axes
body
we now suppose
to turn the contrary
this relative
the same as before.
w' be the angle the system of lines L,
Then, with the plane (x\
,
and making
at rest,
as revolving round it but this line immovable and the
just
O
(
f being the initial value of
We body A
-ST
obtain
z),
we
shall
A
L\
revolves.
&c. forms
have
as before stated, being the angle included by the axes and will be
Moreover the general values of
V= ft (x and the
cos m'
-I-
initial condition, in
plicable, will evidently
twixt
y' sin
and
X, X'.
V
and
')
?= J7r + iy -
Jr',
may be apa quantity be-
order that our solution
become
c
OT O
=
r'
=
TT.
As an example,
let
tan
7
= ,
since
we know by experiment
y is generally very small then taking the most unfavourable case, viz. where VT^ = 0, and supposing the body to make one revolution only, the value of determined from its initial that
one,
;
5J>
= JTT + Jy
JOT'^ will
be found extremely small and only 6
APPLICATION OF THE PRECEDING RESULTS, &C.
82
equal to a unit in the 27th decimal place.
We
thus see with
what rapidity f decreases, and consequently, the body approaches to a permanent state, defined by the equation
Hence, the polarity induced by the rotation is ultimately directed ', along a line, making an angle equal to \TT + 7 with the axis
X
which agrees with what was shown in the former part of
this
article.
The value
of
V
at the
body's surface being thus
known
at
any instant whatever, that of the potential function at a point p exterior to the body, together with the forces acting there, will be immediately determined as before.
APPLICATION OF THE PRELIMINARY RESULTS TO THE THEORY OF MAGNETISM.
The electric fluid appears to pass freely from one part (14.) of a continuous conductor to another, but this is by no means the case with the magnetic fluid, even with respect to those bodies which, from their instantly returning to a natural state the moment the forces inducing a magnetic one are removed, must be considered, in a certain sense, as perfect conductors of mag-
Coulomb, I believe, was the first who proposed to conformed of an infinite number of particles, each of which conducts the magnetic fluid in its interior with perfect freedom, but which are so constituted that it is impossible there shall be any communication of it from one particle to the next. This hypothesis is now generally adopted by philosophers, and netism.
sider these as
its
consequences, as far as they have hitherto been developed,
are found to agree with observation ; we will therefore admit it in what follows, and endeavour thence to deduce, mathema-
the laws of the distribution of magnetism in bodies of any shape whatever.
tically,
Firstly, let us endeavour to determine the value of the pofrom the magnetic state induced in a
tential function, arising
very small body A, by the action of constant forces directed the body being composed of an parallel to a given right line ;
number
of particles, all perfect conductors of magnetism and originally in a natural state. In order to deduce this more immediately from Art. 6, we will conceive these forces to arise infinite
62
'
APPLICATION OF THE PRELIMINARY RESULTS
84 from an
point p on this
an
line, at
of magnetic fluid, concentrated in a Then the infinite distance from A.
of the rectangular co-ordinates being anywhere within be those of the point p, and x, y, z', those of any
origin
A,
Q
infinite quantity
if x, y, z,
other exterior point p, to which the potential function belongs, we shall have (vide Mc. Cel. Liv. 3)
from
V arising
A
1
^(x + y* +
r
z'*)
being the distance Op.
A
is Moreover, since the total quantity of magnetic fluid in (0} = 0. equal to zero, Supposing now r' very great compared
U
jj()
with the dimensions of the body,
all
the terms after
^- in the
expression just given will be exceedingly small compared with this,
by neglecting them,
therefore,
and substituting
for Z7 (1) its
most general value, we obtain
-4,
B, C, being quantities independent of
may
contain x, y,
Now change
a?',
?/',
z',
but which
z.
Fwill remain unaltered, when we and reciprocally. Therefore
(Art. 6) the value of
x, y, z, into x',
y
',
z',
Ax + By' + Cz _ Ax + B'y + G'z ~' B
r*
r'
A,
B', C'j being the
are of
a?,
y, z.
same functions
Hence
it
is
of
a?',
y z, as A, B, C V must be of the t
7
easy to see that
form
v_ d'xx+l"yy'+ c"zz'+e"(xy'+yx) +f"(xz'+zx'}+g"(yz'+zy) rV' a", b"j c", e",f", g",
3
being constant quantities.
If X, Yj Z, represent the forces arising from the magnetism concentrated in p, in the directions of x, y, z, positive, we shall
have
-Qx Y =--
.
-Qy Y = ~~
.
~7 = -Qz.
TO THE THEORY OF MAGNETISM.
V is
and therefore
85
of the form
a'Xx'+V Yy'+c'Zz'+e'(Xy'+ Yx'}+f(Xz'+Zx}+g( Yz'+Zy'}
~>
3
r'
a, lj &c. being other constant quantities. But it will always be possible to determine the situation of three rectangular axes, so that e, f, and g may each be equal to zero, and consequently ',
V be
a, b,
reduced to the following simple form
and
c
being three constant quantities.
When A
is
a sphere, and
magnetic particles are either
its
the
integrant particles of non-crystallized in a confused bodies, arranged manner; it is evident the constant quantities a', &', c', &c. in the general value of F, must be spherical,
like
or,
the same for every system of rectangular co-ordinates, and con-
sequently
we must have a =b' =
c,
e
= o,
f=
o,
and g
= o,
therefore in this case
a'^+Yy' + Zz')
W'|
/3
a being a constant quantity dependant on the magnitude and nature of A.
The formula
give the value of the forces acting on of soft iron or other similar any point p' arising from a* mass matter, whose magnetic state is induced by the influence of the (a) will
A
t
earth's action supposing the distance Ap' to be great compared with the dimensions of A, and if it be a solid of revolution, one of the rectangular axes, say X, must coincide with the axis of ;
revolution,
and the value of
V reduce
itself to
a and V being two constant quantities dependant on the form and nature of the body. Moreover the forces acting in the directions of x, y', z, positive, are expressed by _
fdV\
_ (dV\
_
/
dV
APPLICATION OF THE PRELIMINARY RESULTS
86
We
have thus the means of comparing theory with experiment, but these are details into which our limits will not permit us to enter.
The formula
which
(b),
is
strictly correct for
an
infinitely
small sphere, on the supposition of its magnetic particles being arranged in a confused manner, will, in fact, form the basis of
our theory, and although the preceding analysis seems suffiand rigorous, it may not be amiss to give a ciently general simpler proof of this particular case. Let, therefore, the origin of the rectangular co-ordinates be placed at the centre of the
OB
A, and
be the direction of the parallel since the total quantity of magnetic forces acting upon then, ; the value of the potential function V, to is equal fluid in zero, at the point p arising from A, must evidently be of the form
infinitely small sphere it
A
f
,
T7_ ^ COS ^
-/*-; the angle formed r representing as before the distance Op', and If now fixed in A. between the line Op and another line
OD
,
be the magnitude of the force directed along OB, the constant k will evidently be of the form k = a'f-, a being a constant
f
The
value of F, just given, holds good for any arrangement, regular or irregular, of the magnetic particles comwould posing A, but on the latter supposition, the value of
quantity.
F
evidently remain unchanged, provided the sphere, and consequently the line OD, revolved round OB as an axis, which
could not be the case unless
6
= angle BOp'
OB
and
OD
coincided.
Hence
and
g/C080 r
Let now
12
be the angles that the line Op the axes of x, y, z, and a', ft', 7', those which a, ft, 7,
the same axes
;
then, substituting for cos 6
cos a cos
we
a'
/cosa=JT, /cos/3=Y, '
^
OB
value
+ cos ft cos ft' + cos 7 cos 7',
have, since
v- a
its
=r
cos a
+
^ cos P
makes with makes with
TO THE THEORY OF MAGNETISM.
Which
agrees with the equation
=x
cos a
r
Conceive
(15.)
seeing that
(b),
r
/
cos
,
now
~ y p = ^7 r
87
,
cos
7
=z
.
r
a body A, of any form, to have a magby the influence of exterior
netic state induced in its particles
dv be an element of its volume, the value of the potential function arising from this element, at any point p whose co-ordinates are x, y ', z, must, since the total quantity of magnetic fluid in dv is equal to zero, be of the form forces, it is clear that if
being the co-ordinates of dv, r the distance p, dv and -XT, Y, Z, three quantities dependant on the magnetic state induced in dv, and serving to define this state. If therefore dv x, y, z,
be an
A
and inclosing volume within the body the potential function arising from the whole
infinitely small
the point
p',
A
exterior to dv, will be expressed
by
^ the integral extending over the whole volume of
A
exterior
to dv'. It is easy to
show from
expression that, in general,
this
although dv' be infinitely small, the forces acting in its interior vary in magnitude and direction by passing from one part of it to another ; but, when dv' is spherical, these forces are sensibly constant in magnitude and direction, and consequently, in this case, the value of the potential function induced in dv' by their action,
Let dv' is
be immediately deduced from the preceding article. represent the value of the integral just given, when
may ty'
an
infinitely small sphere.
The
dx'J>
on p' arising x', will be
force acting
from the mass exterior to dv, tending to increase
APPLICATION OF THE PRELIMINARY RESULTS
88
the line above the differential coefficient indicating that it is to be obtained by supposing the radius of dv to vanish after differentiation, and this may differ from the one obtained by first
making the radius
vanish, and afterwards differentiating the y z which last being represented as
resulting function of x,
usual
the
,
,
by -~r we have
first
>
A
exintegral being taken over the whole volume of dv. and the second over the whole of including
A
terior to dv'j
Hence
the last integral comprehending the volume of the spherical particle dv' only, whose radius a is supposed to vanish after
In order to
effect the integration here indicated, that X, and are sensibly constant within and therefore be and their values dv, , may replaced by at the centre of the sphere dv, whose co-ordinates are x y z \
differentiation.
Y
we may remark
Z
X Y
Z
t ,
t
t
t
,
the required integral will thus become
Making
E = Xx + Y y + Z
moment
for a
t
X-
t
Y-
z,
we
shall
Z-
and as also ,1
x
,
x
,1
dr
,1
d-
,
y
y
T
d-
,
z
z
r
~~^^~
have
t
,
t
TO THE THEORY OF MAGNETISM. this integral
may
be written
/
A
dE + -ydxdydz \ -j \ ax ax ay ,
[
,
{
dE
-7-
= 0,
and S -
SE= 0,
which since
dr
r
.
J
89
.
+
d-\
dE r-
.
dz
ay
r)
-7- /
dz
,
/
by what
reduces itself
is
proved
in Art. 3, to
fdE\ -= \dwj
[daI
=
}
j r
.,
(because
,
7 = dw
[da-
N
da)'
] r
dE 7da
;
the integral extending over the whole surface of the sphere dv, of which da is an element ; r being the distance p', da, and
dw measured from -
I
the value of the potential function for a point -j- expresses
p, within with
the sphere, supposing
electricity
obtained
moment
whose density
by No.
13, Liv. 3,
its
is
surface everywhere covered
-y-
Mec.
In
Celeste.
4-
t
a
(X
cos 6
t
being the value of
-
da and as
using for a
fact,
the notation there employed, supposing the origin of the
E= E t
easily be
and may very
,
\JLCb
we have
polar co-ordinates at the centre of the sphere,
E
Now
the surface towards the interior of dv.
+ Y sin
+ Z sin 6 sin t
E at the centre of the sphere.
= X. cos + Y of the form
this is
cos vr
t
&
sin
cos
isr
U w (Vide
+ Z.
Mec.
sin
t*r)
;
Hence
sin
-or,
Celeste, Liv. 3),
we
immediately obtain 7- = 47T/ r da
where /, &, x'j y' and z [da I
dE j~
\X
cos
&+Y
'
sin 0' cos
+Z
'
"
(
v
l^-/
/
v*
>
~~
X\
/J
T ,
\r
f
Jt
(y
,
>
\
y,)
+ .
&
sin
Or by
are the polar co-ordinates of p'.
J 77
sin
rr
^
r
v^
>
'},
restoring
M ~^/)l
APPLICATION OF THE PRELIMINARY RESULTS
90
Hence we deduce
d
titf
d
If
successively
do-
dE
now we make
X'j the value of
the radius a vanish, X must become equal X at the pointy', and there will result t
_ But
'dx~dx'
'
dx'~
dx'
~-T expresses the value of the
x
to
_
force acting in the
on a point p within the infinitely small sphere dv, arising from the whole of A exterior to dv; sub-
direction of
stituting
force
now
positive,
~cW for
its
-p
value just found, the expression of this
becomes
**-% Supposing
V
;:
to represent the value of the potential function at
p, arising from the exterior bodies which induce the magnetic state of A, the force due to them acting in the same direction, is
dT dx"
and therefore the
total
tending to induce a
dv,
the direction of x' positive, state in the spherical element
force in
magnetic
is 4
7rX
In the same way, the positive, acting
upon
dp dV T -j-r=X.
-r-i
total forces in the directions of y' arid z'
dv', are
dV
dx
dx
shown ,
to
be
^,
dty'
"~dV =
TO THE THEORY OF MAGNETISM. 1
the equation (I ) of the preceding article, dv is a perfect conductor of magnetism, and
By when
91
we
see that
its particles
are not regularly arranged, the value of the potential function at any point p", arising from the magnetic state induced in dv the action of the forces X, Y, Z, is of the form
by
a (Xcos
n.
+ Fcos /3 + ^cos 7)
being the distance p", dv', and a, /3, 7 the angles which r' forms with the axes of the rectangular co-ordinates. If then
r
x",
y
',
z" be the co-ordinates of p", this becomes,
that here a
= Jcdv',
kdv'
\X(x" - a?')
+
Y(y" -y]
by observing
+ ~3F (*"-*)}
k being a constant quantity dependant on the nature of the body. The same potential function will evidently be obtained from the expression (a) of this article, by changing dv, p and their co-ordinates, into dv'j p", and their co-ordinates; thus we have ',
dv'{X'(x"-x'}+Y(y"-y'}+Z(z"-z'}} 3
r'
Equating these two forms of the same quantity, there three following equations
results the
:
dx
'
'
dy Zj
==
K^i ==
dy
d^f -k'TrlC^J
rC
5
7~
dz
since the quantities x", y", z" are perfectly arbitrary. ing the first of these equations by dx, the second
third
by dz, and taking
=
(1
their
- frrk) (X'dx +
sum, Y'dy'
we
Multiplythe
by dy,
obtain
+ Z'dz') + k'd+' +
UV
.
APPLICATION OF THE PRELIMINARY RESULTS
92
But dy and dV being perfect differentials, X'dx'+ Y'dy' + Z'dz must be so likewise, making therefore
d$ = X'dx + Tdy + Z'dz', the above,
by
integration, const.
becomes
= (1 - ITT&) fi + fop + kV.
Although the value of k depends wholly on the nature of the body under consideration, and is to be determined for each by experiment, we may yet assign the limits between which it must fall. For we have, in this theory, supposed the body composed of conducting particles, separated by intervals absolutely impervious to the magnetic fluid ; it is therefore clear the magnetic state induced in the infinitely small sphere dv', cannot be greater
than that which would be induced, supposing it one continuous conducting mass, but may be made less in any proportion, at will,
by augmenting
the non-conducting intervals.
When dv is a continuous conductor, it is easy to see the value of the potential function at the point p" 9 arising from the magnetic state induced in it by the action of the forces X, Y, Z, will be
Bdv
X (x
- x'} + Y (y" - y) + Z (z" - z)
=a
a representing, as
seeing that sphere dv.
By
3
-
comparing
this
before, the radius of the
expression with
that before
was not a continuous conductor, it is evident k found, must be between the limits and f TT, or, which is the same thing,
when
g being any The
dv'
positive quantity less than
value of
1.
found, being substituted in the equation serving to determine <', there arises k, just
TO THE THEORY OF MAGNETISM.
93
Moreover
Y= ,,
(ax
I
X(x'-x}+Y(y'-y}Z(z'-z] ay dz 3
A
I
= (dxd dzi^ '
'
'
\dx
J
+& dx
j
*
l ,7
_1 +
'
dy
dy
IV
^ -I) dz '
dz
I
the triple integrals extending over the whole volume of A, and its surface, of which da- is an element ;
that relative to da- over
d the quantities therefore,
by
(/>
and
\
-* Cf/10
belonging to this element.
We
have,
substitution
Now8'F' = 0, and
d\ .dw
and consequently
S'^>'
=
;
the symbol
the co-ordinates of ^/; or, since
making them equal
a?',
y'
S'
and
referring to x, y ', s' a' are arbitrary, by
to x, y, z respectively, there results
0-fc in virtue of which, the value of
r being the distance
p,
i^',
do; and
(
by
Article 3, becomes
-yH belonging
former equation serving to determine 0' gives,
x,
y', z'
to da.
The
by changing
into x, y, s, const.
=
(1
- a] 6 + -$-
(*b
+
V]
.,
..(:
APPLICATION OF THE PRELIMINARY RESULTS
94
V
and , i/r belonging to a point p, within the body, whose coordinates are x, y, z. It is moreover evident from what precedes = &/>, that the functions $, ty and satisfy the equations = &Jr and = BV, and have no singular values in the interior
V
of
A.
The
and -^, comequations (b) and (c) serve to determine from the the value of exterior bodies is arising pletely, enable us to and therefore the known, assign they magnetic state <
V
when
of every part of the body A, seeing that it depends on X, Y, Z, the diiferential co-efficients of <j>. It is also evident that -v/r', when
any point p', not contained within the body A, the value of the potential function at this point arising from the magnetic state induced in A, and therefore this function is calculated for
is
always given by the equation
(&).
The
constant quantity #, which, enters into our formulas, depends on the nature of the body solely, and, in a subsequent article, its value is determined for a cylindric wire used by
This value
Coulomb. therefore
g=
1,
differs
the equations
const.
very (Z)
little
and
(c)
=^r+ V
from unity
:
supposing
become
...................... (c'),
evidently the same, in effect, as would be obtained by considering the magnetic fluid at liberty to move from one part of the
conducting body to another
by
-] (
,
;
the density p being here replaced
and since the value of the potential function
for
any
point exterior to the body is, on either supposition, given by the formula (), the exterior actions will be precisely the same in both cases. Hence, when we employ iron, nickel, or similar bodies, in which the value of g is nearly equal to 1, the observed phenomena will differ little from those produced on the latter hypothesis, except when one of their dimensions is very small compared with the others, in which case the results of the two hypotheses differ widely, as will be seen in some of the applications
which
follow.
TO THE THEORY OF MAGNETISM.
95
If the magnetic particles composing the body perfect conductors, but indued with a coercive force,
were not it is
clear
might always be equilibrium, provided 'the magnetic state of the element dv was such as would be induced by the forces ~ ~ there
d -j-r ax
~d^' --TT
dx
+
dV A d dV B and ~d^' dV Cn instead ofc + + -j-r -r-r + -f-r + -7-7 4-^ ax dz dz ay dy dV d& dV and d& -~ dV the resultant .,
,,,
+ -J-T dx
,
+ -j~r
~-T7
i
.
,
,
,
dy
dz
dy
B
of the forces A', to
,
',
1
+ -yr dz
; '
supposing
C' no where exceeds a quantity
,
measure the coercive
force.
This
is
/3, serving expressed by the con-
dition
the equation
A, B,
C
x, y
a'
',
(c)
would then be replaced by
being any functions of x, y, subject only
z,
as
A
',
B', C'
are of
to the condition just given.
would be extremely easy so to modify the preceding theory, as to adapt it to a body whose magnetic particles are It
regularly arranged,by using the equation (a) in the place of the equation (5) of the preceding article ; but, as observation has not
yet offered any thing which would indicate a regular arrangement of magnetic particles, in any body hitherto examined, it seems superfluous to introduce this degree of generality, ticularly as the omission may be so easily supplied.
more par-
As an application of the general theory contained in (16.) the preceding article, suppose the body to be a hollow spherical shell of uniform thickness, the radius of whose inner surface is a,
A
and let the forces inducing a magfrom any bodies whatever, situate at will, within or without the shell. Then since in the interior of A's mass = 80, and = 8 F, we shall have (Mec. CeL Liv. 3) and that of
its
netic state in
arise
= 2<& V + 2
d>
outer one a/,
A,
c
i
1
and
F=
APPLICATION OF THE PRELIMINARY RESULTS
96
r being the distance of the point p, from the shell's centre, &c., (0)
to
<
which
U
(1)
,
,
(0)
,U
<
(l
\
and V belong, &c. functions of
6 and OT, the two other polar co-ordinates of j?, whose nature has been fully explained by Laplace in the work just cited the to i = co finite integrals extending from i = ;
.
(5)
If now, to prevent ambiguity, we enclose the r of equation Art. 15 in a parenthesis, it will become 'da-
(a
the distance p, do; and the integral extending (r) representing over both surfaces of the shell. At the inner surface we have
r==a: nence tne P art
f
the integrals extending over the whole of the inner surface, and da- being one of its elements. Effecting the integrations by the formulae of Laplace (Mec. C&este, Liv. 3), the part ^, due to the inner surface, viz.
In the same way the part of observing that for
The sum
of these
it
=
-
dw
-fdr
i/r
due
and r
two expressions
=
1
(l-g) 2r+-
+ (1 -ff) 2<
obtain
to the outer surface,
=a
l ,
is
by
found to be
the complete value of ^, before given, being and
is
which, together with the values of < substituted in the equation (c) Art. 15, const.
we immediately
(i)
V
we
rl +
obtain
S U r-^+ (i)
t
Z7
TO THE THEORY OF MAGNETISM.
97
Equating the coefficients of like powers of the variable have generally, whatever i may be,
r,
we
neglecting the constant on the right side of the equation in r as (0) If superfluous, since it may always be made to enter into $ .
now,
we
for
abridgment,
shall obtain
by
we make
elimination
_ 4?r
Z> aHa
A(i)
= _M fTfr(2t'+l)a
D
47T
tt
4-7T
3 47T
'
These values substituted in the expression
give the general value of in a series of the powers of r, when the potential function due to the bodies inducing a magnetic state in the shell is
known, and thence we may determine the
value of the potential function ty arising from the shell for any point whatever, either within or without it.
When shell,
we
itself,
the bodies are situate in the space exterior to the may obtain the total actions exerted on a magnetic all
particle in its exterior, by the following simple cable to hollow shells of any shape and thickness.
.The equation
(c)
Art. 15 becomes,
by
method, appli-
neglecting the super-
fluous constant,
If
now
(<) represent the value of the potential function, cor-
the value of <j> at the inner surface of the shell, responding to each of the functions (<), -^ and F, will satisfy the equations 7 <
APPLICATION OF THE PRELIMINARY RESULTS
98
=
= &/r
= SF,
and moreover, have no singular values in the space within the shell; the same may therefore be said of the function (<),
and as
Q
and
this function is
follows (Art. 5) that
equal to zero at the inner surface, it so for any point p of the interior
is
Hence
space.
But
it
+V
the value of the total potential function at the point p, arising from the exterior bodies and shell itself: this function will therefore be expressed by ijr
is
In precisely the same way, the value of the total potential function at any point p' exterior to the shell, when the inducing 9
bodies are all within
it, is
shown
to
be
being the potential function corresponding to the value of at the exterior surface of the shell. Having thus the total potential functions, the total action exerted
in
on a magnetic particle
immediately given by differentiation. To apply this general solution to our spherical shell, the inducing bodies being all exterior to it, we must first determine <>,
any
direction, is
the value of
<
at its inner surface,
making
= %U}
i}
r~
i
~l
since
there are no interior bodies, and thence deduce the value of (<). and <, their values before given, making Substituting for < (i)
=
and r
= a, we
(i)
obtain
and the corresponding value of
()
is
(Mec. CeL Liv, 3)
TO THE THEORY OF MAGNETISM.
The value
of the total potential function at
the shell, whose polar co-ordinates are
99
any point p within
r, 0, VT, is
In a similar way, the value of the same function at a point exterior to the shell, all the inducing bodies being within
p'
it, is
found to be
r,
6 and
OT in this
expression representing the polar co-ordinates
of/.
To give a very simple example of the use of the first of these formulas, suppose it were required to determine the total action exerted in the interior of a hollow spherical shell, by the magnetic influence of the earth ; then making the axis of x to coincide with the direction of the dipping needle, and designating by f, the constant force tending to impel a particle of positive fluid in the direction of x positive, the potential function V, to the exterior bodies, will here become
due
V= -/. x = -/cos O.r = U *.r. The finite integrals expressing the value of V reduce themselves (
which i=l, and the
therefore, in this case, to a single term, in
corresponding value of
D being
the total potential function within the shell
is
1
-(!-,)
We
therefore see that the effect produced shell, is to reduce the directive force which
by the intervening would
act
on a very
small magnetic needle,
from
/,
to
1+ f-
72
APPLICATION OF THE PRELIMINARY RESULTS
100
In iron and other similar bodies, g
is
very nearly equal to
1,
and
therefore the directive force in the interior of a hollow spherical shell is greatly diminished, except when its thickness is
very small compared with its radius, in which case, as is evident from the formula, it approaches towards the original value /,
and becomes equal
to
it
when
this thickness is infinitely small.
To give an example of the use of the second formula, let it be proposed to determine the total action upon a point p, situate on one side of an infinitely extended plate of uniform thickness, when
another point P, containing a unit of positive fluid, is placed on the other side of the same plate considering it as a
For this, let fall the perpenperfect conductor of magnetism. the dicular the side of prolonged, plate next P, on upon
PQ
PQ
demit the perpendicular pq, and make
and
= the
PQ = 5,
Pq =
u,
pq =
v,
thickness of the plate then, since its action is evidently equal to that of an infinite sphere of the same thickness, at an infinite distance from P, whose centre is upon the line t
;
QP
we
_
have the required value of the total potential function at p by supposing a = a + 1, a infinite, and the line PQ prolonged to be the axis from which the angle 6 is measured. shall
t
Now
__
in the present case
TT_ "
_J_
~
2
-2r(a-
\
b) cos
+ (a - 6)
_ V 77
(i)*.-*-1
2
}
and the value of the potential function, as before determined,
first expression we see that the ~l 1 a quantity of the order (a - I} r~ substituting for r its value in u,
From
U
-+-*
r
the
general
is
term
l
is
.
Moreover,
by
compared with neglecting such quantities as are of the order
those retained. (i)
Z7,
,
The
general term CT/'V"*'
1 ,
and consequently
as functions of ought therefore to be considered
= 7.
TO THE THEORY OF MAGNETISM. In the
finite integrals just
101
given, the increment of
the corresponding increment of
7
-
is
= dy
* is 1,
(because a
is
and in-
the finite integrals thus change themselves into ordinary In fact (Mec. C6L Liv. 3), or fluents. always satisfies the equation
finite),
U
integrals
and as 6
small whenever V has a sensible value, from the above by means of the equation
is infinitely
we may
eliminate
ad
and we obtain by neglecting
= v,
it
infinitesimals of higher
f\
orders than those retained, since -
dv
Hence the value
U
]
2
is
= 7,
vdv of the form
seeing that the remaining part of the general integral becomes when v vanishes, and ought therefore to be rejected. It
infinite
now only remains to determine the value of the arbitrary con= 0, i.e. v = 0, we stant A. Making, for this purpose, have
Ur = (a- by
By
substituting for
Z7/V" =
^^
and
(
A
and r
= 4*
:
hence (a-bj
their values, there results
- 5)' (a - 6 + )^
cos
APPLICATION OF THE PRELIMINARY RESULTS
102
because Cl
=7
and -
= dy.
Writing now in the place of
CL
value ay, and neglecting infinitesimal quantities,
i its
we have
4
D Hence the value of the
total potential function
becomes
where the integral relative to 7 is taken from 7 = to 7 = GO and co of i, seeing that i = ay. to correspond with the limits
,
The preceding
solution is immediately applicable to the case imaginary only, in which the inducing bodies reduce themselves to a single point P, but by the following simple artifice we may give it a much greater 'degree of generality :
Conceive another point P', on the line PQ, at an arbitrary distance c from P, and suppose the unit of positive fluid con-
P; then if we make r = Pp, and have u = r' cos 0', v = r sin 0', and the
centrated in P' instead of
&
*.
pPQ, we
shall
value of the potential function arising from P' will be
Moreover, the value of the total potential function at p due and the plate itself, will evidently this, arising from
P
to
be obtained by changing u into u is
Expanding c,
c in that before given,
and
therefore
this function in
the term multiplied
by
an ascending
c* is
series of the
powers of
TO THE THEORY OF MAGNETISM. which, as c
term
(i}
Q -^
bodies.
is
103
must be the part due
perfectly arbitrary,
to the
in the potential function arising from the inducing
If then this function
V
"
jf
V^ ya
had been
V
where the successive powers c,
V
/a
r 1
r
/
4
2
&c. of c are replaced by the arbitrary constant quantities k & x &2 &c., the corresponding value of the total potential function will be given by making c
,
c ,
a like change in that due to P',
,
,
Hence
,
if,
for
abridgement,
we
make fc
\7 + r4^7
3
s
+&c.
)
the value of this function at the point p will be
P
the original one due to the point be called F, clear the expression just given may be written
Now,
if
it is
where the symbols of operation are separated from those of quantity, according to ARBOG AST'S method thus all the difficulty ;
is
reduced to the determination of F.
Kesummg magnetic
state
therefore the original supposition of the plate's being induced by a particle of positive fluid con^-
centrated in P, the value of the total potential function at will be
as
was before shown. Let
= m we :
shall
have
p
APPLICATION OF THE PRELIMINARY RESULTS
104
^= 2
'
1
i ""
_m)
(i
'o
Writing now
we
provided
**
f
.
e""^ 1"^"
1
shall
~
J-
imaginary quantities which
obtain
may
arise.
this double integral let
have
o~.
I
~
I
I
T7^
,Q2\
1
"~o
I
/
the integral relative to z being taken from z
The
we
in the place of cos (fiyv),
reject the
In order to transform
and we
cos
1
=
to
i
2
=
)
.
2+^
#, for iron and other similar bodies, is very therefore small; neglecting quantities which are of the order those with retained, there results (1 g) compared
value of
1
'
where u and v may have any values whatever provided they
105
TO THE THEORY OF MAGNETISM.
and of the order
are not very great
F becomes by changing u into u +
^
If
.
F
represents
t
& 2t,
what
'
we have
^r-!.W^F)L^
st
'*""
;
and consequently
which, by effecting the integrations and rejecting the imaginary quantities,
becomes
Suppose now
is
j?
a perpendicular
falling*
from the point
p
surface of the plate, and on this line, indefinitely extended in the direction Op, take the points p^ p# pa , &c., at the
upon the
distances 2, 4,
6^,
&c. from p
;
then
Fv F F z,
values of F, calculated for the points
p^ p# p
mula
3,
(a) of this article,
values of
r',
we
and
shall equally
r'
l9
r'
2
,
r'
B,
s,
&c. being the &c. by the for-
&c. the corresponding
have
and consequently
F= | (1 - g) (, + seeing that
From
-
+ -*- + &c.
in
;
infinitum)
J^ = 0.
this value of F, it is evident the total action exerted
upon the point p, in any given direction pn, is equal to the sum of the actions which would be exerted without the interposition of the plate, on each of the points p,
p^
pv p# &c. in infinitum, n &c. multiplied by the constant pz z
in the directions
pn
factor - (1
the lines pn, pji^
g)
:
Moreover, as this
t
is
,
p^ &c. being
all parallel.
the case wherever the inducing point
P
APPLICATION OF THE PRELIMINARY RESULTS
106
good when, instead of P, we whatever body any figure magnetized at will. The only condition to be observed, is, that the distance between p and every part of the inducing body be not a very great quan-
may be
situate, the
substitute a
same
tity of the order
On
.
when
the contrary,
inducing body
will hold
of
is
the
distance between
great enough to render
(1 t
p
and the
a)r' ^J a very con-
F
in will be easy to show, by expanding a descending series of the powers of /, that the actions exerted upon are very nearly the same as if no plate were interposed.
siderable quantity,
it
p
We have before remarked (art. of a
body
15), that
when
the dimensions
same order, the results of from those, which would be obtained
are all quantities of the
the true theory differ little by supposing the magnetic like the electric fluid, at liberty to move from one part of a conducting body to another; but when,
as in the present example, one of the dimensions is very small compared with the others, the case is widely different; for if we
make g
rigorously equal to 1 in the preceding formulae, they will belong to the latter supposition (art. 15), and as .Fwill then vanish, the interposing plate will exactly neutralize the action
of any magnetic bodies however they may be situate, provided they are on the side opposite the attracted point. This differs
completely from what has been deduced above by employing like difference between the results of the the correct theory.
A
two suppositions takes place, when we consider the action exerted by the earth on a magnetic particle, placed in the interior of a hollow spherical shell, provided its thickness is very small compared with its radius, as will be evident by making g 1 in
the formulas belonging to this case, which are given in a preceding part of the present article. 17.
Since COULOMB'S experiments on cylindric wires magnet-
ized to saturation are numerous and very accurate, it was thought this little work could not be better terminated, than by directly
deducing from theory such consequences as would admit of an immediate comparison with them, and in order to effect this, we
TO THE THEORY OF MAGNETISM.
107
suppose a cylindric wire whose radius is a exposed to the action of a constant force, equal
will, in the first place,
and length 2X, is and directed
to f,
parallel
to the
axis of the wire, and then
endeavour to determine the magnetic state which will thus be induced in it. For this, let r be a perpendicular falling from a point
p
within the wire upon
and
its axis,
x, the distance of
the foot of this perpendicular from the middle of the axis; then being directed along x positive, we shall have for the value of
f
the potential function due to the exterior forces
and the equations
(V), (c)
(art. 15)
become, by omitting the su-
perfluous constant,
the distance^', do- being inclosed in a parenthesis to prevent ambiguity, and j/ being the point to which ^r' belongs. By the (r),
= S and = 8^r, and as cf> and ty evir x and on only, these equations being written at dently depend same
article
we have
length are
Since r
is
always very small compared with the length of the in an ascending series of the powers of /*,
wire, we may expand
and thus
X,
Xv Xv
etc.
being functions of x only.
value in the equation just given, of like powers r, we obtain
*-z-^^+^ a+ 9~
dx* 2
By
substituting this
and comparing the
*' 2
2
dx* 2 .4
+ etC
-
coefficients
APPLICATION OF THE PRELIMINARY RESULTS
108
In precisely the same way the value of ty
It
now only remains
tion
By (c)
found
to
be
-
=
of x.
is
to find the values of
supposing p
X and Y in functions
placed on the axis of the wire, the equa-
becomes
F=
~JMWW
;
the integral being extended over the whole surface of the wire : Y' belonging to the point p\ whose co-ordinates will be marked
with an accent.
The
Y' due to the = X, is x where cylinder,
dX"
a
[
= %7rrdr 2
of the order a
the value of
end of the
dX"
27rrdr
71
since here da-
At
circular plane at the
part of
and
~ = -^
/7V""
by
,
neglecting quantities
on account of their smallness
X when x =
;
X"
representing
X.
the other end where
+ X we
x
dX'" dx
d$ = dw and consequently the part due
to
have
da-
= 2irrdr
9
J
it is
dX' X'" designating the value
At
of
X when x = + X.
the curve surface of the cylinder
= da = %7radx and.3$ -f dw ,
provided retained.
we omit
d$ = l d*X -f\a j-sdr dx2
quantities of the order
Hence the remaining ,
Tra
,
a2 compared with those
part due to this surface
d*X [dx -r^:
I
J (r)
j-s-
dx*
; '
is
TO THE THEORY OF MAGNETISM. the integral being taken from value of Y' is therefore
\
x
109
x= + \.
to
The
total
2 V X - x' + a + X - a'
27T
2
2
a _2,r^[y{(X + a0 + }--X-
_
2
Cdx 'dxd z
X
J(r) the limits of the integral being the same as before. If now substitute for (r) its value *J{(x we shall have + x'Y af]
dxd*X both integrals extending from
On
x=
account of the smallness of
d*X
dx
f
we
\tox = + \. a,
the elements of the last in-
tegral where x is nearly equal to x are very great compared with the others, and therefore the approximate value of the expression
just given, will be * A iraA
d*X' where 2 j-fi dx ,
A= .
f /
^
- +}
J *J{(x
dx
^-
t
x)
^
= Ol 2 log
2/4
a
p and + p, very nearly ; the two limits of the integral being and p so chosen that when p is situate anywhere on the wire's immediate vicinity of either end, the approxifrom the true value, which may in without done be case difficulty. Having thus, by substievery Y' free from of the a value tution, sign of integration, the value axis, except in the
mate
of
shall differ very little
Y is given by merely
this
changing x into x and
way
dx ,
.
X
'
into
X
;
in
APPLICATION OF THE PRELIMINARY RESULTS
110
The
or
by
equation
(c),
substituting for
by making
X
to
a;
= 0,
becomes
Y
an equation which ought
#=
r
hold good, for every value of #, from
to
= + X. -
In those cases to which our theory will be applied, 1 g is a small quantity of the same order as a?A, and thus the three terms of the first line of our equation will be of the order a*AX;
making now x = a*AX'",
order
aA
+ X, f g
dX'" ,
a
is
and therefore^-:- X'" ax ;
shown is
to
be of the order
a small quantity of the
but for any other value of x the function multiplying
J~V~i" 7 ax
2 becomes of the order a and therefore we
sensible
,
error
neglect the term
containing
it,
may
without
and likewise
suppose
dx In the same way by making x =
X, it
term containing
and
J'V"
is j
negligible,
dx Thus our equation reduces
itself to
may
be shown that the
Ill
TO THE THEORY OF MAGNETISM. of which the general integral
where
2
/3
=
^
7f
Determining -r
-^
>
OOCL JL
is
an ^
^
the
conditions
being two arbitrary constants.
JV
these
by
-r -X"'"
obtain efto-e-P*
(
3gf
\
VP)| But the density of the would produce the same
dw
the magnetized wire z
dr
dx*
2
and therefore the total quantity in an whose breadth is dx, will be ,
Tra -j-g-
dx
dx
=
--
-j-ff
4(1
which
itself, is
X
d = ~dd> ^L = _ _ a __ very
s
f
fluid at the surface of the wire, effect as
I
-
and
^
X" we ultimately -
fff
=
c
^)
.
infinitely thin section
-fi A
e^
nearly,
-
-,
zr dx.
+ e-p^
As the constant quantity may represent the coercive force of steel or other similar matter, provided we are allowed to suppose this force the same for every particle of the mass, it is clear
f
that
when
a wire
is
magnetized to saturation, the effort it makes be just equal to
to return to a natural state must, in every part,
f\ and
on account of its elongated form, the degree of retained magnetism by it will be equal to that which would be in a induced conducting wire of the same form by the force f, therefore,
directed along lines parallel to its axis. Hence the preceding But it has formulas are applicable to magnetized steel wires.
been shown by M. BIOT (Traitt de Phy. Tome 3, Chap. 6) from COULOMB'S experiments, that the apparent quantity of free fluid in
any
infinitely thin section is represented
by
This expression agrees precisely with the one before deduced
APPLICATION OP THE PRELIMINARY RESULTS
112
from theory, and gives, and fjf, the equations
for the determination of the constants
A'
The
chapter in which these experiments are related, contains also a number of results, relative to the forces with which magnetized wires tend to turn towards the meridian,
from
and
when
retained
easy to prove that this force for a fine wire, whose variable section is s, will be proporat a given angle
it,
is
it
tional to the quantity / I
i
sdx
u(p 7
dx
j
where the wire
is
magnetized in any
way
otherwise, the integral extending over
its
either to saturation or
whole length.
But in
a cylindric wire magnetized to saturation, we have, by neglecting quantities of the order d
dX
= ~f dx dx -jand
a2
,
Zgf A
4-7T
-* *
7T^
^~-
~
f
an "
s
=
^
(1
therefore for this wire the force in question (
is
proportional to
"
The
value of #, dependent on the nature of the substance of which the needles are formed, being supposed given as it ought to be, we have only to determine ft in order to compare this result
But
with observation.
depends upon
on account of the smallness of
a,
A
A = 2 log
undergoes but
,
and
little altera-
tion for very considerable variations in /u-, so that we shall be able in every case to judge with sufficient accuracy what value of p,
ought to be employed
:
nevertheless, as
to avoid every thing at all vague,
A
by the
condition, that the
sum
it
it is
always desirable
will be better to determine
of the squares of the errors
committed by employing, as we have done,
A d*X' ,
,
2
for the ap-
TO THE THEORY OF MAGNETISM.
-
.
proximate value of
shall be a minimum for 2 +<*} In this way I find when X is so
-777 J-W{(-aO ,
the whole length of the wire.
V2
.
,
great that quantities of the order -^pA,
,
may
be neglected,
A = .231863 - 2 log a/3 + 2aj3 where .231863
&c. = 2
log 2
-
113
2 (A)
;
(A)
;
being the quantity
represented by A in LACUOIX' Traite da Cal. Diff. Tome 3, p. Substituting the value of A just found in the equation
before given,
we
obtain (1
(*)
t
6g
We
521.
.
"tf2 a
p
= -231863 - 2 log a/3 + 2a.
hence see that when the nature of the substance of which
the wires are formed remains unchanged, the quantity a/3 is This constant, and therefore /3 varies in the inverse ratio of a. agrees with what M. BIOT has found by experiment in the chapter before cited, as will be evident by recollecting that
From an experiment made with extreme on a magnetized wire whose radius was found the value of p' to be .5 1 7948 Hence we have in this case a/3
=
log
fju
(
care
by COULOMB,
inch,
M. BIOT has
TraitS de Phy.
Tome 3,
p. 78).
= .0548235,
which, according to a remark just made, ought to serve for all steel wires. Substituting this value in the equation (a) of the present article,
we
obtain
g
With
this value of
= .986636.
g we may
different lengths of a steel wire
calculate the forces with
whose radius
is
which
inch, tend to
114
APPLICATION OF THE PRELIMINARY RESULTS
turn towards the meridian, in order to compare the results with the table of COULOMB'S observations, given by M. BIOT (Traitt de Phy.
Tome
force for
have before proved that this
any wire may be represented by
where, for abridgment,
It
Now we
3, p. 84).
we have supposed
has also been shown that for any steel wire aj3
= .0548235,
the French inch being the unit of space, and as in the present case a
=
,
there results
fore to determine
from which
we
/3
= .657882.
It only remains there-
K from one observation, the obtain
K= 5 8. 5 very nearly
first for ;
example,
the forces being this value of
measured by their equivalent torsions. With we have calculated the last column of the following table
The
last three observations
K
:
have been purposely omitted, be(a) does not hold good for very
cause the approximate equation short wires.
The very small difference existing between the observed and calculated results will appear the more remarkable, if we reflect that the value of /3 was determined from an experiment of quite a different kind to any of the present series, and that only one of these has been employed for the determination of the constant
TO THE THEORY OF MAGNETISM.
115
quantity K, which depends on f, the measure of the coercive force.
The
table page 87 of the volume just cited, contains another of observed torsions, for different lengths of a much finer 1 / 38 wire whose radius a */ - hence we find the correspondset
=
ing value of
/3
=
3,1
gives ^T=.6448.
:
3880, and the first observation in the table these values the last column of the
With
following table has been calculated as before
Here
also the differences
:
between the observed and calculated
values are extremely small, and as the wire is a very fine one, our formula is applicable to much shorter pieces than in the former case. In general, when the length of the wire exceeds
10 or 15 times
its
diameter,
we may employ
it
without hesita-
tion.
82
MATHEMATICAL INVESTIGATIONS CONCEKNING THE
LAWS OF THE EQUILIBRIUM OF FLUIDS ANALOGOUS TO THE ELECTRIC FLUID, WITH
OTHER SIMILAR RESEARCHES*.
From
the Transactions of the Cambridge Philosophical Society, 1833.
[Read Nov.
12,
1832.]
MATHEMATICAL INVESTIGATIONS CONCERNING THE LAWS OF THE EQUILIBRIUM OF FLUIDS ANALOGOUS TO THE ELECTRIC FLUID, WITH OTHER SIMILAR RESEARCHES.
AMONGST
the various subjects which have at different times occupied the attention of Mathematicians, there are probably
few more interesting in themselves, or which offer greater difficulties in their investigation, than those in which it is required determine mathematically the laws of the equilibrium or motion of a system composed of an infinite number of free particles all acting upon each other mutually, and according to to
some given law. When we conceive, moreover, the law of the mutual action of the particles to be such that the forces which emanate from them may become insensible at sensible distances, the researches to which the consideration of these forces lead will be greatly simplified by the limitation thus introduced, and may be regarded as forming a class distinct from the rest.
Indeed they then for the most part terminate in the resolution of equations between the values of certain functions at any point taken at will in the interior of the system, and "the values of the
When
same point. in question continue sensible at every finite distance, the researches dependent upon them become far partial differentials of these functions at the
on the contrary the forces
more complicated, and often require all the resources of the modern analysis for their successful prosecution. It would be easy so to exhibit the theories of the equilibrium and motion of ordinary fluids, as to offer instances of researches appertaining to the former class, whilst the mathematical investigations to
which the theories of Electricity and Magnetism have given may be considered as interesting examples of such as belong
rise
to the latter class.
ON THE LAWS OF
120 It is not
my
chief design in this paper to determine mathe-
matically the density of the electric fluid in bodies under given circumstances, having elsewhere* given some general methods by which this may be effected, and applied these methods to a variety of cases not before submitted to calculation. present object will be to determine the laws of the equilibrium of an
My
hypothetical fluid analogous to the electric fluid, but of which the law of the repulsion of the particles, instead of being inversely as the square of the distance, shall be inversely as any power n of the distance; and I shall have more particularly in view the determination of the density of this fluid in the interior of conducting spheres when in equilibrium, and acted upon by any exterior bodies whatever, though since the general method by which this is effected will be equally applicable to circular plates and ellipsoids. I shall present a sketch of these applications also.
It is well known that in enquiries of a nature similar to the one about to engage our attention, it is always advantageous to avoid the direct consideration of the various forces acting upon
p
of the fluid in the system, by introducing a parany particle ticular function of the co-ordinates of this particle, from the
V
differentials of
which the values of
diately deducedt. endeavoured, in the
We first
all
these forces
may
be imme-
have, therefore, in the present paper place, to find the value of F, where the
density of the fluid in the interior of a sphere is given by means of a very simple consideration, which in a great measure obviates the difficulties usually attendant on researches of this kind, have to determine the value F, where p, the density of the
been able
any element dv of the sphere's volume, is equal to the product of two factors, one of which is a very simple function fluid in
*
Essay on
tricity
the
Application of Mathematical Analysis
to the
Theories of Elec-
and Magnetism.
f Tiiis function in the present case will be obtained by taking the sum of all the molecules of a fluid acting upon p, divided by the (w-l) th power of their respective distances from p; and indeed the function which Laplace has represented by V in the third book of the Mecanique Celeste, is only a particular value of our more general one produced by writing 2 in the place of the general ex-
ponent
n.
THE EQUILIBRIUM OF FLUIDS. containing an arbitrary exponent
/?,
121
and the remaining one
f is
equal to any rational and entire function whatever of the rectangular co-ordinates of the element dv, and afterwards by a
proper determination of the exponent /3, have reduced the resulting quantity Fto a rational and entire function of the rectangular co-ordinates of the particle p, of the f.
This being done,
it is
same degree
as the function
easy to perceive that the resolution of
the inverse problem may readily be effected, because the coefficients of the required factor will then be determined from the
f
given coefficients of the rational and entire function F, by means of linear algebraic equations.
The method in the
two
alluded to in what precedes, and which is exposed of the following paper, will enable us to
first articles
assign generally the value of the induced density p for any ellipsoid, whatever its axes may be, provided the inducing forces are
given explicitly in functions of the co-ordinates of p but when by supposing these axes equal we reduce the ellipsoid to a ;
sphere,
it is
natural to expect that as the form of the solid has
become more simple, a corresponding degree of simplicity will be introduced into the results and accordingly, as will be seen in the fourth and fifth articles, the complete solutions both of the direct and inverse problems, considered under their most general ;
point of view, are such that the required quantities are there always expressed by simple and explicit functions of the known ones, independent of the resolution of
The
first
five
any equations whatever.
articles of the present
analytical, serve to exhibit the relations
paper being entirely exist between the
which
density p of our hypothetical fluid, and its dependent function F; but in the following ones our principal object has been to point out some particular applications of these general relations.
In the seventh
article, for
example, the law of the density of
when
in equilibrium in the interior of a conductory been has investigated, and the analytical value of p there sphere,
our fluid
found admits of the following simple enunciation. The density p of free fluid at any point p within a conducting is the centre, is always proportional to the sphere A, of which th radius of the circle formed by the interthe of (n 4) power
ON THE LAWS OF
122
section of a plane perpendicular to the ray Op with the surface of the sphere itself, provided n is greater than 2. When on the n is less than 2, this law requires a certain modification ; contrary
the nature of which has been fully investigated in the article just named, and the one immediately following.
been remarked, that the generality of our analysis will enable us to assign the density of the free fluid which would be induced in a sphere by the action of exterior forces, It has before
supposing these forces are given explicitly in functions of the rectangular co-ordinates of the point of space to which they But, as in the particular case in which our formulae belong.
admit of an application to natural phenomena, the forces in question arise from electric fluid diffused in the inducing bodies,
we have
in the ninth article considered more especially the case of a conducting sphere acted upon by the fluid contained in any exterior bodies whatever, and have ultimately been able to exhibit the value of the induced density under a very simple
form, whatever the given density of the fluid in these bodies
may
be.
The
tenth and last article contains an application
general method
to circular planes,
from which
results,
of the
analogous
for spheres in some of the preceding ones, are and towards the latter part, a very simple formula is given, which serves to express the value of the density of the free fluid in an infinitely thin plate, supposing it acted upon by
formed
to those
deduced
;
other fluid, distributed according to any given law in its own Now it is clear, that if to the general exponent n we plane.
assign the particular value 2, all our results will become appliIn this way the density of the cable to electrical phenomena. thin circular plate, when under the electric fluid on an infinitely influence of
any
electrified bodies
whatever, situated in
its
own
The analytical expression which plane, will become known. serves to represent the value of this density, is remarkable for its simplicity ; and by suppressing the term due to the exterior bodies, immediately gives the density of the electric fluid on a circular conducting plate, when quite free from all extraneous action.
Fortunately, the manner in
which the
electric
fluid
THE EQUILIBRIUM OF FLUIDS. distributes itself in the latter case, has long since
123 been deter-
mined experimentally by Coulomb. We have thus had the advantage of comparing our theoretical results with those of a very accurate observer, and the differences between them are not be supposed due to the unavoidable errors of experiment, and to that which would necessarily be produced by
greater than
may
employing plates of a finite thickness, whilst the theory supposes this thickness infinitely small. Moreover, the errors are all of the same kind with regard to sign, as would arise from the latter cause. 1.
of
If
we
conceive a fluid analogous to the electric
fluid,
but
which the law of the repulsion of the
particles instead of being inversely as the square of the distance is inversely as some power n of the distance, and suppose p to represent the density
of this fluid, so that dv being an element of the
A
through which
volume of a
diffused, pdv may represent the contained in this element, and if afterwards we write g quantity for the distance between dv and any particle p under considera-
body
tion,
it
is
and these form the quantity
" v _pdv, the integral extending over the whole volume of A, it is well known that the force with which a particle p of this fluid situate in
any point of space is impelled in the direction of any line $ and tending to increase this line will always be represented by 1
1
n \dq
V
being regarded as a function of three rectangular co-ordinates of p, one of which co-ordinates coincides with the line ^, and
being the partial differential of F, relative to this
last co-
ordinate.
In order
now
to
make known
the success of our general
mainly depends, simple example.
it
will
method
the principal artifices on which for determining the function
be convenient
F
to
begin with a very
ON THE LAWS OF
124
Let us therefore suppose that the body
A
is a sphere, whose the origin of the co-ordinates, the radius being 1 ; and p is such a function of x, y\ z , that where we substitute for x, y' 9 z their values in polar co-ordinates
centre
is at
r
x =/ it
cos
0',
r sin
y
ff
cos
OT',
z'
= r'
sin
&
sin
OT',
shall reduce itself to the form
being the characteristic of any rational and entire function whatever which is in fact equivalent to supposing
f
:
Now, when
*
as in the present case, p can be expanded in a powers of the quantities a?', y, z', and of the will always various products of these powers, the function series of the entire
V
admit of a similar expansion in the entire powers and products of the quantities x, y, z, provided the point p continues within the body A*, and as moreover evidently depends on the distance Op = r and is independent of 6 and w, the two other
V
polar co-ordinates of p,
we
it is
easy to see that the quantity V,
when
substitute for x, y, z these values
x=r will
cos 0,
y
= r sin 6 cos OT,
become a function of
r,
z
= r sin 6 sin
-or,
only containing none but the even
powers of this variable.
But
since
we have
dv^r'dr'dffdvsuiO',
and p
= (1 -r *)*./(/),
the value of F becomes
= f P^ = /r' *
The
2
dr'
d& d*'
sin
&
truth of this assertion will become tolerably clear,
if
we
recollect that
V
regarded as the sum of every element pdv of the body's mass divided by the (n - l) th power of the distance of each element from the point p, supposing the density of the body A to be expressed by p, a continuous function of x, y , z. For
may be
then the quantity V is represented by a continuous function, so long as p remains within A ; but there is in general a violation of the law of continuity whenever the point p passes from the interior to the exterior space. This truth, however, as enunciated in the text, is demonstrable, but since the present paper is a long one, J have suppressed the demonstrations to save room.
THE EQUILIBRIUM OF FLUIDS. the integrals being taken from to 6'
TT,
and from
Now V may
V
/=
OT'
/=
to
=
to
-57'
125
= 27r,
&=
from
1.
be considered as composed of two parts, one
B
whose centre is at the origin 0, and surface passes through the point p, and another V" due to the In order to obtain the first part, we must shell S exterior to B. l ~" in an ascending series of the powers the expand quantity g due
of
In
.
gl-n
to the sphere
this
way we
= |yj _ 2rr
If then
we
get
+ Q m Q gm Q> CQS '
>
J
cog Q CQS
fl'
substitute this series for
1
g
^
^ -)}+
l-n 2
2
r' ]
in the value of F', 2
3 T*' we effect the )* having expanded the quantity (1 integrations relative to /, 0', and -or we shall have a result of
and
after
,
',
the form
V = r*-" {A + Br + W + &c.} z
V
before represented by seeing that in obtaining the part of the integral relative to r ought to be taken from r' to r
V,
=
=r
only.
To ^
1 ~n
in
obtain the value of F",
an ascending
series of the
we must expand powers of
,
the quantity
and we
shall thus
have l ~n
g
= (r*
l-n
2rX
the coefficients
[cos
$
,
6 cos
& + sin 6 sin 6' cos (r
^, $2 &c. being ,
of
+ r'
2
* )
the same as before.
The expansion arise
')]
here given being substituted in F", there will a series of the form
which the general term T, =Jd0' thi sin
ff
T
8
is
QJr" dr'
(1
-
ON THE LAWS OF
126
to r' = l, from 0' = to This 2?r. will be evident by integral by which the value of V" is
the integrals being taken from & = 7r, and from v?' = Q to OT' =
r=r
recollecting that the triple expressed, is the same as the one before given for V, except that to the integration, relative to r, instead of extending from r'
=
r
= 1,
ought only to extend from r
But the general term
by
A
2t t
r'
,
the part of
Ar t
jdV a
T
r to r
in the function f(r'
dependent on
8
sin
= I.
6'.
QJr'-*-
n
this dr' (1
2
) being represented term will evidently be
- r'^
(2)
;
the limits of the integrals being the same as before,
We thus see that the value of T8 and consequently of V" would immediately be obtained, provided we had the value of the general integral
JV'*-'(t-T. which being expanded and integrated becomes i
but since the
first
of this expression
line
is
the well-known
expansion of
r nj
)
or
\nj
nr
<1>
when n = 2.p = b+l and
W
q
=2
(/3
+ 1) we
have ultimately,
By means of the result here obtained, we shall readily find the value of the expression (2), which will evidently contain one term multiplied by r8 and an infinite number of others, in all of
THE EQUILIBRIUM OF FLUIDS.
127
which the quantity r is affected with the exponent n. But, as in the case under consideration, n may represent any number whatever, fractionary or irrational, it is clear that none of the terms last mentioned can enter into F, seeing that it ought to contain the even powers of r only, thence the terms of this kind entering into ones in V.
V" must necessarily be destroyed by corresponding By rejecting them, therefore, the formula (2) will
become
(2').
s
But
-
V ought
even powers of r only, those odd number, will vanish an which the exponent of themselves after all the integrations have been effected, and as
to contain the
terms in
s is
consequently the only terms which can appear in
V
t
are of the
form
(4);
where, since
s is
an even number, we have written
2s' in
the
place of 5, and as Q& is always a rational and entire function of cos 0', sin & cos /, and sin ff sin OT', the remaining integrations
may
immediately be
effected.
Having thus the part of T'& due
any term
to
function f(r*) we have immediately the value of T' quently of V" 9 since
V"=U'+T '+TJ+ 9
2y +
Ar t
-
tat
of the
and conse-
T + &c.; f
e
V representing the sum of
all the terms in V" which have been of their account on form, and T T^ T the value of rejected T TI T^ &c. obtained by employing the truncated formula (2) in the place of the complete one (2). '
But or
by
- F= V'+ V" =V' + U'+ T '+
Zy +
TJ+ T^
transposition,
7_ zy _ Tj - Tl -
T;
- &e. =
V + U';
ON THE LAWS OP
128
and as in this equation, the function on the left side contains none but the even powers of the indeterminate quantity r, whilst that on the right does not contain any of the even powers of r, it that each of
is clear
In
zero.
this
way
its
the
ought to be equated separately
sides
left side
to
gives
F=r;+2y+7y + r;+&c
(5).
Hitherto the value of the exponent /3 has remained quite arbitrary, but the known properties of the function T will enable us so to determine
/3,
that the series just given shall contain
We
number of terms only. the value of F, and reduce it a
finite
function of r
For
(0)
and entire
2 .
this purpose,
r
shall thus greatly simplify
in fact to a rational
= oo r ,
If therefore
we may remark
(-
1)
that
= oo r (- 2) = oo m mfinitum. ,
we make
,
h /3
=
any whole number
positive
or negative, the denominator of the function (4) will become infiand consequently the function itself will vanish when s is
nite,
7?
-+ /3 + + 3
so great that
t
s'
is
2i
number, and as the value of
equal to zero or any negative
never exceeds a certain num-
t
2
ber, seeing that f(r' ) is a rational and entire function, it is clear that the series (4) will terminate of itself, and become a
V
rational 2.
article,
and
entire function of
r*.
The method that has been employed in the preceding where the function by which the density is expressed is
of the particular form
p-(l-OW), may, by means of a very more general value p
= (1 - r'
where
/ is
slight modification, be applied to the far
2 ) "/(*',
y',
)
= (1 - a? -y" -
")'/V,
*',
<0,
the characteristic of any rational and entire function reduces F to a /3 which
whatever: and the same value of rational
and
entire function of
2
r in the
first case,
reduces
it
in
THE EQUILIBRIUM OP FLUIDS.
129
the second to a similar function of x, y, z and the rectangular co-ordinates of p.
To
prove
we may remark
this,
that the corresponding value
V will become
F=//WJ6W sin & (1 - /y/(V,
y', z'}
f
;
the integral being conceived to comprehend the whole volume of
the sphere.
Let now the function /be divided into two /(*', y'
z'}
,
=/
(*', y', z')
parts, so that
+/ (*', y',
z)
;
containing all the terms of the function/, in which the sum ot the exponents of x', y' y z is an odd number ; and/ the remaining In this terms, or those where the same sum is an even number.
/
way we
get
F=F1+ F
2
the functions
F and F
2
t
;
corresponding
to/ and/, being
F =fr dr'd0'dvr' sin (I - r") Vi <X> #'> F -fMdffdJ sin tf (1 - /*) V, (a/, /, 2
We will and
^
x
*')
2
O /-
in the
first
F
place endeavour to determine the value t by writing for x, y z their values before ;
f
for this purpose,
given in /,
we
0', OT',
,
get
z
2
the coefficients of the various powers of r' in i|r (r ) being evidently rational and entire functions of cos ff, sin & cos OT', and sin0' sintn-.
Thus
- r'
'
sin this
integral,
like
the
(1
foregoing,
2
)?
/^
/2
(r )
^-
;
comprehending the whole
volume of the sphere.
Now
as the density corresponding to the function
V
l
is
in an ascending series of the y, z, and the various products of these powers consequently, as was before remarked (Art. 1), F, ad9
it is
clear that
entire
it
powers of
may be expanded a?',
THE LAWS OF
130
mits of an analogous expansion in entire powers and products of Moreover, as the density p^ retains the same numerical x, y, z.
and merely changes its sign when we pass from the element dv to a point diametrically opposite, where the co-ordi-
value,
f
nates
z are replaced
#', y',
F
that the function
1?
by
x,
y depending upon p lt ',
easy to see possesses a similar
z': it is
property, and merely changes its sign when x, y, z, the co-orz. Hence the nature dinates of p, are changed into x, y, is such that it can contain none but the of the function t
V
odd powers of ordinates x, y,
when we
r,
substitute for the
z,
co-
rectangular
their values in the polar co-ordinates
r, 0, TV.
V
is Having premised these remarks, let us now suppose divided into two parts, one F/ due to the sphere which passes through the particle p, and the other V" due to the exterior .
1
B
shell 8.
p=
(1
-
Then
it is
/2
2
7*
)0/(r
the coefficients
A,
by proceeding,
as in the case
where
that F; will be of the form
),
= r*-n {A + Br* +
F; variable
evident
JB,
(7,
O
4
+ &c.}
;
&c. being quantities independent of the
r.
In like manner we have also F/'
= fr'Wdffdw
sin
&
(1
-/
=r
the integrals being taken from r 0'
= TT,
and from
-&'
=
f
to vr
2
)0.
/^
to
(V
2 )
f*
;
r=l, from & =
to
= 2?r.
1-n before used substituting now the second expansion of # (Art. 1), the last expression will become
By
F "=r + r +2
7
1
1
of which series the general term
T =fdffdrf sin ff Q a
8
2
+r +& c 3
.
is
fr'*-dr' (1
-/
/.
2 Moreover, the general term of the function ty (r' ) being represented by A r*\ the. portion of T8 due to this term will be t
sn the limits of the integrals being the same as before.
THE EQUILIBRIUM OF FLUIDS.
M
now we
If
the formula
131
by means
effect the integrations relative to r
and
of
reject as before those
powers of the variable r, in which it is affected, with the exponent n, since these ought not to enter into the function FI} the last formula will
Art.
(3),
1,
become
^-V(+i) sin0'Q8 A
r
8
r fd0'dv
and as T^ ought
make
s
= 2s
(a'),
t
none but the odd powers of r, we may and disregard all those terms in which 5 is an
to contain
-f 1,
even number, since they will necessarily vanish after all the Thus the only remaining terms operations have been effected. will be of the form
r
/4-. + g .-.*rx
,::,-,,,;
.
;
2. 2
& 0' =
A
and Q +l are both rational and entire functions of sin & sin-cr', the remaining integrations from & cos & = TT, and CT' = to *r' = 2?r, may easily be effected in
where, as cos
,
t
zs
sin
,
to
}
-sr',
the ordinary way. If article,
now we
follow the process
7 and suppose Zy, T /,
T^
employed
in the preceding
&c. are what
T T T ,
I}
z
,
&c.
become when we use the truncated formula (a) instead of the complete one
(a),
we
shall readily get
In like manner, from the value of "
F ^pWdffd*' sin ff 2
the integrals being taken from r 0'
= TT,
and from
OT =
Expanding now g
to l
~n
(1
V
%
before given,
- r"*Y $ (r'
=r
to
we
get
2
)/-"
;
r=l, from
^'
=
OT = 2?r.
as before,
we have
92
to
THE LAWS OP
132
*
where 1
U = jdd'd'&
sin tr
a
f
Q
r'
1
8
J
U
and the part of
3 ~n
dr
f
r8
2
r'
(1
)^
r
due to the general term
B
2 <
(r' ),
g
V
By*
in
2
(/
(b)
;
),
will be
r Jdffdvr sin
^' (28
5
f
t
v'
r
'^~ n+2t ~ g
dr
- r'
(1
2
)^
r
which, by employing the formula
(3')
Art.
1,
and rejecting the
inadmissible terms, gives for truncated formula '
(
+
1}
continuing to follow exactly the same process as was beemployed in finding the value of F^, we shall see that s
By fore
must always be an even number, say 2s' sion immediately preceding will become
;
and thus the expres-
-2^ 2
\
/
Moreover, the value of
*%>
we
Ui> U*> Us>
&c
use the formula
The
value of
by adding
F
-
V
2
will be
bein ^ what
(b'}
U U U >
i>
*>
-
(b).
answering to the density
together the two parts into which
divided, therefore,
&c Become when
instead of the complete one
becomes
it
was
originally
133
THE EQUILIBRIUM OF FLUIDS.
When
taken arbitrarily, the two series entering into extend in infinitum, but by supposing as before, Art 1,
ft>
ft
is
representing any whole number, positive or negative,
it is
V
clear
from the form of the quantities entering into T# +l and Z72 ,,, and from the known properties of the function F, that both these series will terminate of themselves, and the value of V be expressed in a finite form; which, by what has preceded, must necessarily reduce itself to a rational and entire function of the rectangular co-ordinates x, y, z. has before been advanced (Art.
we
will, therefore,
It
only remark that
of the function /(a/, ascend will be
y
,
+
2o>
co
positive or negative ft
=
7
any
which
V can
may ;
but
represent any whole if
we make
co
=
4-
77
,
:
represents the degree
+ 4.
In what immediately precedes,
and consequently,
if
what
after
proof of this
z), the highest degree to
7
number whatever,
seems needless,
1), to offer
the degree of the function
2,
V is
2i
the same as that of the factor
comprised in
This factor then being supposed the most gene-
p.
ral of its kind, contains as
many arbitrary constant quantities as there are terms in the resulting function V. If, therefore, the form of the rational and entire function be taken at will, the
V
arbitrary quantities contained in /(a?', y, z) will in case always enable us to assign the corresponding value of p,
co
=
.
2
and the
resulting value of f(x, y z'} will be a rational and entire function of the same degree as V. Therefore, in the case now under ,
consideration,
of
V
when p
we is
be able to determine the value also have the means of solvshall but given, shall not only
ing the inverse problem, or of determining p when V is given ; and this determination will depend upon the resolution of a certain
number of algebraical
equations, all of the
first
degree.
THE LAWS OF
134
The
3.
object of the preceding sketch has not been to point way of finding the value of the function
out the most convenient V, but
merely
make known
to
the spirit of the method
and
;
to
success depends. Moreover, when presented in this simple form, it has the advantage of being, with a very modification, as applicable to any ellipsoid whatever as to
show on what
its
slight
But when spheres only are to be considered, the sphere itself. the resulting formulae, as we shall afterwards show, will be much more simple
if
we expand the density p in a series by Laplace (Mec. Gel. Liv.
similar to those used
of functions :
iii.)
it
will
however be advantageous previously to demonstrate a general property of functions of this kind, which will not only serve to simplify the determination of V, but also admit of various other applications of do;
Y
is a function of and r, of the form Suppose, therefore, considered by Laplace (Mec. Gel. Liv. iii.), r, 6, v? being the fixed in space, polar co-ordinates referred to the axes X, Y", (i)
Z
t
so that
x = r cos then, if
we
0,
y
r sin 6 cos
-or,
z
= r sin
6 sin
conceive three other fixed axes JT15
the same origin but different tion of #x and -zzTj, and may
Y
directions,
therefore be
(i)
will
cr
;
Y Z t
,
11
having
become a func-
expanded
in a series of
the form
Suppose now we take any other point p and mark co-ordinates with an accent, in order to distinguish those of p\ then, shall
if
we
designate the distance
pp by
its
various
them from (p, p'),
we
have
= [r - 2rr 2
L.^
o)
as has been
(cos
+
cos
&+
Qd^r +
^
shown by Laplace
where the nature of the completely explained.
sin
"I*
r
sin 0' cos
+
0)
r
_l r
in the third
(w -*')}
+ r'T*
+ &c \ )
book of the Mec. Gel,
different functions here
employed
is
THE EQUILIBRIUM OF FLUIDS.
the same quantity is expressed in the belonging to the new system of axes
In like manner,
if
co-ordinates
polar JTj,
Y
and
it
135
Z^ we have, since the quantities r and r are evidently l the same for both systems, ,
i
may
series just
represent,
changing
from the form of the radical quantity of given are expansions, that whatever number will be immediately deduced from Q by
also evident
is
which the
0,
w,
Q
ff, is
(i}
into
VF 19 #/, 1?
w/.
But
since the quan-
9*
is
tity
indeterminate, and
equating the two values of
be taken at
may
we
will,
--^ and comparing the
.
get,
like
by
powers
v of the indeterminate quantity <2
If
now we
,
=
multiply the equation
(6)
by
the element of a
h (h) spherical surface whose radius is unity, and then by Q = Q^ \ we shall have, by integrating and extending the integration over the whole of this spherical surface,
fdfjidv
Which
Q
(h)
Y^ =fd^d^ QM YW + Y" +
equation,
and
(h}
Q
Y
(i
{
by
the
known
properties
Yf>
+ &c.}.
of the
functions
\ reduces itself to
when h and
i represent different whole numbers. But by means of a formula given by Laplace (Mec. CeL Liv. iii. No. 17) we may immediately effect the integration here indicated, and there will thus result 4>7r
y(h)
.
~2h + l** being what
Y^
^
becomes by changing 9V into #/, r t ', which belong to p\ may be taken arbitrarily like the first, we shall have generally (h)
Y^ and
as the values of these last co-ordinates,
THE LAWS OF
136
YJ
i. Hence, the expansion except when h a single term, and becomes
h} ,
(6)
reduces
itself to
We
Y
thus see that the function
even when referred the same origin
(i}
continues of the same form
any other system of axes with the first. to
Xv Yv Zv having
This being established,
let us conceive a spherical surface of the co-ordinates and radius /, the whose origin of which with the covered fluid, then, if da' density p = Y' represent any element of this surface, and we afterwards form
center
is at
(i}
;
the quantity
the integral extending over the whole spherical surface, g being the distance p, da-' and ty the characteristic of any function
whatever.
R
being a function of
F'
(i)
0, OT,
V will be
I say, the resulting value of
r,
Op only and
the distance
becomes by changing 6
of the form
what
the like co-ordinates of the point p.
To lt
(i}
/, the polar co-ordinates, into
',
justify this assertion, let there be taken three so that the point p may be upon the axis
X Yv Zv the
Y
new
polar co-ordinates of da'
of ^> being
r, 0,
-cr,
be written
X^
;
then,
r', 0', -or',
those
and consequently, the distance will become
g= and as da
may
new axes
2
2**r'
^/ (r'
cos
2
0j'
+r
)
j
r'^dO^d^^ sin #/, we immediately obtain
F= / F m r "^ r
1
cfer
1
sin
0^ (r - 2rr 2
z
0j'
sin 6^ ty (r
f
cos
0/+ r'
2
)
%rr' cos
Let us here consider more particularly the nature of the integral i
In the preceding part of the present
article,
it
has been
THE EQUILIBRIUM OF FLUIDS. shown
that the value of Y' (i\ ordinates, will be of the form
form (Vide Mec. series containing
Gel. Liv.
137
when expressed in the new cobut all functions of this F/
iii.)
(i)
;
may
be expanded in a
finite
2i+ terms, of which the first is independent and each of the others has for a factor a sine or I
of the angle -cr/, cosine of some entire multiple of this same angle. Hence, the integration relative to OT/ will cause all the last mentioned terms to vanish,
But
this
T
(
and we
term
is
shall only have to attend to the first here. to be of the form
known
i.il
n
H*-2^^
1
,.
and consequently, there 2
f
J
-1
ft|
^
^ >V'd) a if H F =27r H^
i.i
+2
2 .i
1 .i
3
.4. 2 f-i.af-8
\
p lV ~ .
,.
r
will result
i'i- 1
/.-a
2^?=I*
+
*.*'-l.*-2.4"-3
,
,,
2.4.24-1.27^
4
\ p - &C 'j'
where /*/ = cos #/ and A; is a quantity independent of 6t and OT/, but which may contain the co-ordinates 0, -or, that serve to define '
the position of the axis It
X
v
now only remains
passing through the point p. to find the value of the quantity k,
which may be done by making #/ = 0, for then the line r coincides with the axis v and Y during the integration remains
X
constantly equal to
(i}
Y
(i)
,
the value of the density at this axis.
Thus we have rii\ -
>
7
"."
A,
1
or,
the
by summing the
common
series
1.42.4
,4.4
=2^(1-^^+ 2-4
.
2
3
p
\
._ 1-2 ._ 3 -&C.J:
within the parenthesis, and supplying
factor 2?r,
"1.3.5 ......... 2*-! and, by substituting the value of Jc, drawn from this equation in the value of the required integral given above, we ultimately obtain
THE LAWS OF
138
If now, for abridgement, ,.
we
i.il
,.
shall obtain,
found in that of
we make
,;
_4
p
by substituting the value of the integral just
V before
given,
which proves the truth of our
From what
i.i- 1.^
assertion.
has been advanced in the preceding
article, it is
likewise very easy to see that if the density of the fluid within a sphere of any radius be every where represented by
(f)
being the characteristic of any function whatever;
and we
afterwards form the quantity
where dv represents an element of the sphere's volume, and g the distance between dv and any particle p under consideration, the resulting value of V will always be of the form
Y
(i]
being what
Y' w becomes
by changing /} OT', the polar co-ordinates of the element dv into 0, OT, the co-ordinates of the being a function of r, the remaining co-ordinate point p and
E
;
of p, only. 4.
Having thus demonstrated a very general property
functions of the form
Y
(i
\ let
us
now endeavour
of
to determine the
V for
a sphere whose radius is unity, and containing fluid of which the density is every where represented by
value of
y z being the rectangular co-ordinates of dv, an element of the sphere's volume, and/, the characteristic of any rational and entire function whatever. a?',
t
THE EQUILIBRIUM OF FLUIDS. For
we
this purpose
139
will substitute in the place of the co-
ordinates x' y, z their values 9
x =r
cos
6'
r sin
y
:
& cos ix
z'=r
:
sin
& sin <&'
;
and afterwards expand the function f(x' y z) by Laplace's simple method (Mec. Gel. Liv. iii. No. 16). Thus, ',
t
f(x, 5
=/"" +/" +/* + &c ....... +/<
y', z')
being the degree of the function f(x, y It is likewise easy to
/'
and as every
U\ (i
f
(i)
r
r'
=/.'
+/,'
we
fw
of this
r
w +/;
may always
now we can
r"
M+
&c.;
developement
of the form
is
easy to see that the general term reduced to a rational and entire function
it is iii.),
be
of the original co-ordinates x, If
z).
perceive that any term
coefficient of the last
(Mec. Cel. Liv. i+2t
;
be again developed thus,
may
expansion
,
......... (7)
y
',
z'.
obtain the part of
shall immediately
V
due
have the value of
V
to the
term
by summing
all
the
parts corresponding to the various values of which i and t are But from what has before been proved (Art. 3), susceptible.
the part of
form
Y
(i)
'
}
V now under consideration
must necessarily be of the we shall by V
representing, therefore, this part
(i)
t
j
readily get [
sn
J
is
-^r
Moreover from what has been shown in the same we have generally
Y
being the characteristic of any function whatever, and becomes by substituting 0, to- the polar co-ordinates of
what Y'
p
article, it
easy to see that
(i}
(i)
in the place of
0',
-or',
the analogous co-ordinates of the element
THE LAWS OF
140
If therefore in the expression immediately preceding,
dv.
we
make Y'
(i}
'
=f
(i)
t
n~ f (g = g = (ff*)~** z
and
l
)
,
and substitute the value of the integral thus obtained equal in
V
(i)
t
V u= 27T/f I'l'l'"^
1
t
where
f
for (i),
t
(i)
is
V^W J f
deduced from
abridgement, n _ fr-ft~
As
for its
there will arise
*
is
'
1
(1
dn\ (i)
-ryj
'
(i}
t
by changing
0', OT'
into 6, OT,
and
written in the place of the function
i.i
i-s
the integral relative to
/// x
l.i2. i-
3
which enters
into the expression
,<_ 4
on the right side of the equation (8) is a definite one, and depends therefore on the two extreme values of // x only, it is evident that in the determination of this integral, it is altogether useless to But by omitting retain the accents by which p l is affected.
we
these superfluous accents, the quantity
dp
I
shall
have
z
(i)
.
(r
2rr'fjL
to calculate the value of
+ r'
*
2 )
;
J-4
where
i.i-l
,
,
.
i.il.i-V.i-
3
{
The method which first presents itself for determining the value of the integral in question, is to expand the quantity l-n 2
(
r
_ 2rr'fj, + r'
a
2 )
by means of the Binomial Theorem,
to replace
the various powers of p by their values in functions similar to and afterwards to effect the integrations by the formulas (i) For this purpose contained in the third Book of the Mec. CeL
we. have the general equation
THE EQUILIBRIUM OF FLUIDS.
To remove
may
all
141
doubt of the correctness of this equation, we its sides by /u,, and reduce the products on
multiply each of
the right by means of the relation
which form
it is
(i).
very easy to prove exists between functions of the In this way it will be seen that if the equation (9)
holds good for any power p*
ing power
//*"*,
and as
it
is
it
will
therefore necessarily so, whatever
Now by
do so likewise
evidently correct
for the follow-
when
whole number
i
i
=
1,
it is
may represent.
means of the Binomial Theorem, we obtain when
r
= ^ _2
r r
V
z r \\
+r
J
2,
If
conceive the quantity
the same theorem,
by ,
now we
it is
f2//,-,
^J
to
be expanded
easy to perceive that the term having
r\ for factor is
(-, j
i+2^
i+2t
,
r
i+ztf
2.4... n^ '
2.4... &c ......................... &c ...................... &c............. and therefore the function
/y.y+2*-
coefficient of (-,1
<
in the expansion of the
THE LAWS OF
142 will be expressed
by
'--2
4
.
,
1V
......
5-l ...... *+
2'
25-
Hence the portion
of this coefficient containing the function (/), the various powers of //. have been replaced by their values in functions of this kind agreeably to the preceding observation will be found, by means of the equation (9), to be
when
.Ti
/\ 2^-1 ^ ~2
+1
7&
+ 2/+4J' 25-3
2/+4'-2s 1 2.9 i + 2t' 25 / + 2' /+ 1 4. ..2'- 25X2/4- 2*' -25 + 1 2/ + 2*'- 25- 1...2^-f 4
'.
.
x 2.
.
+ 2*'-s.* + 2*'-5-l
iv
oi^-2./
1.2.3
2.4.6 + .
3
1
.
......
i+ 2
.
"+ 3
.
^+4 ...... t + 2^'-
8 (- 1) Ti-1 .n + l.n+3
"' ^ ...
.
3.5.7 ...... 1.2.3 ......
i '
""
^ :2
2.4.6 2.4.6
where
=
all
the finite integrals
=
and
......
may
25
evidently be extended from
clear that the last of these integrals , v in the product the coefficient of x to 'equal .-s
to 5
x
]
co
it is
is
THE EQUILIBRIUM OF FLUIDS.
143
If now we write in the place of the series their the preceding product will become
and consequently the value of the required
7i-2. n
.
+
7i
2 ......... n-\-
,
coefficient of
-
2t'
values,
*
=(!-)
x (l-o?)
(l-x}~
known
xv
is
4 *
~~2
6
74~.
2?
.........
This quantity being substituted in the place of the last of the finite integrals gives for the value of that portion of the coefficient of
which contains the function 3.5.7...2/+1 1.2.3...
By sum
X
n-l .n+l.. 3
t
5
.
,
(i)
the expression
.
2
.
+ 2*'+l
2
multiplying the last expression by
(
of all the resulting values which arise
.4...
,
j
and taking the
when we make
suc-
cessively t'
we
0, 1, 2, 3, 4, 5, 6,
&c. in infinitum,
shall obtain the value of the
Y
term
(i)
contained in the
expression l-n
r
__ 2//,
r
,
+
r2 \
2
= Y + Y + y + Y + &c. (0)
r
)
(2)
(l)
(B}
Hence (i] Y(i]
_ ~ 3.5
...2/+1
n-l. n + l
. ..
...n
+ 2i+2t f
3
1.2...
n-
4 /rV^' "
;
2. 4.. .21
(r'J
the finite integral extending from
But by the known have by substituting
for
t'
=
to
t'
GO
.
properties of functions of this kind,
Y
(i}
its
value
we
144
THE LAWS OF
3.5.7...2ft
+
1.2.3...
%
r^v "n
2
.
n
3. 5. ..2ft
..
2. 4. ..2*'
=2
1.2.3...
n-l
i
1.3.5...2ft-l
2. 4. ..2
now we
(Ifec.
(7e?.
Liv.
iii.
No. 17)
/1. 2. 3...
2
If
+2' + 2
ST.
f
by what Laplace has shown
since
restore to the accents with which it was origiand multiply the resulting quantity by r'"" 1 we //,
nally affected, shall
.n 3. 5.. ..2ft
,
have when r
1
r l
l
J-i
1.2.3...
,
*
1
.
3
.
5
... 2ft
+ - 1 gn-l.n 3.5...
+ + 2*'-3 + 2*' + 1
l...n
ft
W-2.W
...
w
2ft
2ft
+ 2'-4
2. 4. ..2*'
and
in order to deduce the value of the
we shall only have to change r into formula just given.
We
may now
the formula
it
(8).
follows from
same integral when / < r, r', and reciprocally, in the
F
4 by means of readily obtain the value of For the density corresponding thereto being
what has been observed
present article, that f^r**
9*
may always
(i
in the former part of the be reduced to a rational
THE EQUILIBRIUM OF FLUIDS. and entire function of x', y, z the rectangular co-ordinates of element dv, and therefore the density in question will
the
admit of being expanded in a series of the entire powers of x, y, z and the various products of these powers. Hence admits of a similar expansion in entire powers &c. (Art. 1) V i]
t
of x, y, z the rectangular co-ordinates of the point p, and by following the methods before exposed Art. 1 and 2, we readily
get
2.4.6...
and thence we have ultimately,
2.4... It
^ y
+ 2-^
...
.
w-2.w...w + 2^n
1
.
n
+1
.
.
.
2.4...
n + 2i + 2*' - 3
oo and T being to t' the finite integrals being taken from t' = the well known function of the characteristic Gamma, which is
introduced when of the formula
(3),
Having thus //
(0 ,
we
Art.
V
in /(#', y' 9 z)
effect the integrations relative to r
t
(i)
by means
1.
or the part of
V
corresponding to the term the complete value
we immediately deduce
10
THE LAWS OF
146 of
V by
giving to i and
t
the various values of which these
susceptible, and taking the sura of all the parts corresponding to the different terms in the expansion of the func-
numbers are tion f(x' 9 y,
z').
in the present Article we have hitherto supposed of a rational and entire function, the characteristic the to be
f
Although
same process will evidently be applicable, provided /(#', y' z) can be expanded in an infinite series of the entire powers of In the latter z and the various products of these powers. cc', y the case we have as before development 9
',
f(x',
y', z')
=/"! +/"
1
+/"" + &c.
+/
(i of which any term, as for example f' \
may
be farther expanded
as follows,
/*"
and as we have already determined
V
(i)
t
+ &c.
or the part of
V cor-
i+
we immediately deduce as before the responding to f{ >r the of value V, only difference is, that the numbers i required (i
',
and
t,
instead of being as in the former case confined within may here become indefinitely great.
certain limits,
In the foregoing expression if
we
ber,
assign to
it
-
(11) 13
such a value that
may
be taken at
may
but
be a whole num-
the series contained therein will terminate of
V
will,
itself,
and
(i)
consequently the value of t will be exhibited in a finite form, capable by what has been shown at the beginning of the present Article of being converted into a rational x, y, z, the rectangular co-ordinates of p.
and
entire function of
It is
moreover evi-
V
dent, -that the complete value of being composed of a finite number of terms of the form t will possess the same property, which provided the function /(a/, #', z} is rational and entire,
V
(i}
Article agrees with what has been already proved in the second a method. different by very 5.
We have before remarked
case where
fi
=
*M
n_-ii
(Art. 2), that in the particular
A. ,
the
arbitrary
constants
contained in
THE EQUILIBRIUM OF FLUIDS. /(a/, y,
mine
z'}
are just sufficient in
number
147
to enable us to deter-
V
make
the resulting value of equal to any given rational and entire function of x, y, z, the rectangular co-ordinates of p, and have proved that the corresponding this function, so as to
V
same degree. But when this the method there proposed becomes imconsiderable, degree it the that resolution of a system of requires practicable, seeing functions
and f
will be of the
is
8
+1
.S 1
+ 2.S+3
.2.3
linear equations containing as many unknown quantities ; s being But by the aid of what the degree of the functions in question. has been shown in the preceding Article, it will be very easy to the function f(x', y , z) determine for this particular value of 1
and consequently the density p when
V
is
given, and
we
shall
thus be able to exhibit the complete solution of the inverse pro-
blem by means of very simple formulae.
For
let us suppose agreeably to the precedthe that density of the fluid in the element dv p ing remarks, is of the form
this purpose,
/ being the characteristic same degree
as F,
termine, that the value of
any
given rational
of a rational
and which we and
and
entire function of the
will here endeavour so to de-
V thence
resulting,
entire function of
a?,
may
Then by Laplace's simple method (Mec. Gel Liv. we may always expand F in a series of the form
F= F
ill.
s.
No. 16)
(8)
(2)
(0)
be equal to
y, z of the degree
In like manner as has before been remarked, we
shall
have
the analogous expansion f
)
of
=y
which any term,
as follows,
<>
f
(i}
for
example,
may
be farther developed
102
THE LAWS OF
148
'
/
(i)
(l)
>
O
/i'
>
^'
(i) '
the form Y'
(i)
&c
of being quantities independent of r and all V of the V Moreover t part (Mec. Gel Liv. in.). Mt of the last be obwill term series, -
(i}
due to the general n tained
for /3 in
by writing
wards substituting
In
f^r'
~
this
way we
for
get *
*
*?
2.4...2*+2i"-l
sm
n-l.n+1
x
2. 4. ..2*'
f
the equation (11), and after-
being what j^
(i]
t
the finite
becomes by changing 6', to t' integral being taken from t'
Let us now
for a
.
4
. . .
2
(i) .
/J
.r*
into 8,
and
-or,
.
y ^
n
I
.n+1
...
^Bf
n + Si+St'
!
,-k
S
.,-v.f..
5
2t'
then the expression immediately preceding 2-7T
TV
= co
moment assume. 4
2
+ 2i'4-2*'-3
...w
^4
w.6-n...
2^
may
2^'
+2
be written
w
,
T\
t]
w
I'*
sm
and by giving
to t the various values 0, 1, 2, 3, &c. of which it and susceptible, taking the sum of all the resulting values of V the quantity thus obtained will be equal to F or that part of V which is of the form Y . Thus we get
is
(i]
(i)
t
(i}
THE EQUILIBRIUM OF
149
FLUIDS.
sm
......................
&c ...................... &c ....................
F
(i) since all the terms in the preceding value of in f vanish of themselves in consequence of the factor
4
%. 6
n...
2t-
2tf'
+2
which
t'
>
t
n
2.4 ...2t-2t'
(when *'>
But
F
(i)
as deduced from the given value of
Fmay
t).
be expanded
in a series of the form
F"> = r*.
and
if
F + F^V + F (i)
{
2
4
(i) .
r
+ F3
(i)
6
r
+ &c.}
in order to simplify the remaining operations,
we make
generally 2?r
3
n-2.?i...rc
+ 2~-4
2. 4. ..2*
sm
n-l.n+l
...
3. 5.. ,2t 271-2
-2
.
}
'
THE LAWS OF
150
the equation immediately preceding will become
x ft
'
i
>+4>(l).^'V+4>(2)
(0).I7
tf2 >.r<-f &c.)
sm
V
which compared with the foregoing value of
(i
\
will give
by
'
*""
suppressing the factor
,
'_
common
to
both and equat-
different powers of the indeing separately the coefficients of the r the following system of equations terminate quantity A
~
n *S 2.4
4
~
n
4-n.
, (i) '
**
6
n.S
2.4.6
~n
, {i)
J*
"
&c ................................. &c ............................... &c. for
determining the
unknown
U
means of the known ones
Q
functions
(i
\
U,
/
(i)
,
f, f\
Z72 (i) , &c.
U =f
In
&c. by
fact the last
(i} and then by ascending s 8 equation of the system gives successively from the bottom to the top equation, we shall get the values of 8 , /, . 1 j/jw &c. with very little trouble, it will
/
(i)
our equations
,
( i)
,
however be simpler
(i)
,
still to
remark, that the general type of
all
is
where the symbols of operation have been separated from those of quantity and e employed in its usual acceptation, so that
But
it is
evident
we may
satisfy the last equation
Expanding now and replacing eZ7
tt
values
R&, RiS,&c.weget
(i)
,
e
2
Z7tt (i) ,
by making
&c. by these
THE EQUILIBRIUM OP FLUIDS.
151
2.4
2
n
4
.
n
2
.
n
2.4.6
we may immediately deduce fj
from which
and thence suc-
(i}
cessively
2>
f(x', "i
y',
and p
)
= (1
-/">+/' +/'< + &c... +/"", /2
/.a
cc
'2
y
'2\
&
)
*>
_/?
/
'
/
f (m V ?
&
'\
)
Application of the general Methods exposed in the preceding Articles to Spherical conducting Bodies.
In order to explain the phenomena which electrified 6. bodies present, Philosophers have found it advantageous either to adopt the hypothesis of two fluids, the vitreous and resinous of
Dufay
for
example, or to suppose with ^Epinus and others,
that the particles of matter when deprived of their natural quanIt is easy tity of electric fluid, possess a mutual repulsive force. to perceive that the mathematical laws of equilibrium deducible
from these two hypotheses, ought not to
differ
when
the quan-
or fluids (according to the hypothesis we choose to adopt) which bodies in their natural state are supposed to contain, is so great, that a complete decomposition shall never tity of fluid
be effected by any forces to which they
may be exposed, but that in every part of them a farther decomposition shall always be possible by the application of still greater forces. In fact the mathematical theory of electricity merely consists in determining p* the analytical value of the
fluid's density, so% that the
* It
may not be improper to remark that p is always supposed to represent the density of the free fluid, or that which manifests itself by its repulsive force ; and therefore, when the hypothesis of two fluids is employed, the measure of the excess which we choose to consider as positive over that of any element dv of the volume of the body is expressed by pdv, whereas on the other hypothesis pdv serves to measure the excess of the quantity of fluid in the element dv over what it would possess in a natural state. of the quantity of either fluid
the fluid of opposite
name
in
THE LAWS OF
152
whole of the
electrical actions exerted
at will* in the interior of the
upon any pointy, situated conducting bodies may exactly
destroy each other, and consequently p have no tendency to move in any direction. For the electric fluid itself, the ex-
ponent n is equal to 2, and the resulting value of p is always such as not to require that a complete decomposition should take place in the body under consideration, but there are certain values of n for which the resulting values of p will render fpdv greater than any assignable quantity for some portions of the ;
it is
body
therefore evident that
of the fluid or fluids is
supposed
may
be,
how
which
to possess, it will
great soever the quantity
in a natural state this
body
then become impossible strictly
to realize the analytical value of p,
and therefore some modifi-
cation at least will be rendered necessary, by the limit fixed to the quantity of fluid or fluids originally contained in the body,
and as Dufay's hypothesis appears the more natural of the two, shall keep this principally in view, when in what follows it become may requisite to introduce either.
we
The
7.
foregoing general observations being premised,
we
will proceed in the present article to determine mathematically
the law of the density p, when the equilibrium has established itself in the interior of a conducting sphere supposing it free from the actions of exterior bodies, and that the particles of fluid
A
,
contained therein repel each other with forces which vary inth versely as the 7i power of the distance. For this purpose it be may remarked, that the formula (1), Art. 1, immediately gives the values of the forces acting on any particle p, in virtue of the repulsion exerted by the whole of the fluid contained in
A.
In this way we get 1
I __
.
dV =
-7
the force directed parallel to the axis X,
~j~
the force directed parallel to the axis Y,
I
_n 1
dV
_n
~j-
the force directed parallel to the axis Z.
THE EQUILIBRIUM OF FLUIDS. But is
153
consequence of the equilibrium, each of these forces
since, in
equal to zero,
we
shall
have
,, dV dV dx dV7 = -j+ T dy + -^- dz = d V\ ,
dx
and
therefore,
by
,
,
-
,
dz
dy
integration,
V
const.
V
at the point p, whose co-ordiHaving thus the value of nates are x, y> z, we immediately deduce, by the method explained in the fifth article,
m ^- 2
sin
(
= P
A
^
&'n
seeing that in the present case the general expansion of given reduces itself to
V F
If moreover
Q
fluid in the sphere, .
r
.
serve to designate the total quantity of free
we
/n-2
sin
(0)
\ TT
shall have, .
A
its
by
substituting for TT
,
value
n
"" ON 2
See Legendre, Exer. de Cal.
Int.
4
""
nN
,
)
Quatrieme Partie.
In the preceding values, as in the
article cited, the radius
taken for the unit of space ; but the same forreadily be adapted to any other unit by writing
of the sphere
may
..
)r(
r(
mulae
F there
is
i
- in the place of a
r',
and recollecting that the quantities
and Q, are of the dimensions
0,
4
w,
and 3
p,
F,
respectively, with
regard to space a being the number which represents the radius of the sphere when we employ the new unit. In this way we obtain ;
THE LAWS OP
154
w-2 sin
Hence, when Q, the quantity of redundant fluid originally introduced into the sphere is given, the values of Fand of the In fact, by writing in the predensity p are likewise given. ceding equation for ,
\ fn-2 ?rk
.
and sin their values,
we
f
thence immediately deduce
and
F=--
a
- Q.
VTT
formulae present no difficulties where n > 2, the value of p, if extended to the surface of the sphere A, would require an infinite quantity of fluid of one name to have been originally introduced into its interior, and there-
The foregoing
but when n
fore,
< 2,
agreeably to a preceding observation, could not be strictly In order then to determine the modification which in
realized.
ought to be introduced, let us in the first place make n > 2, and conceive an inner sphere B whose radius is a $a, in which the density of the fluid is still defined by the first of
this case
the equations (12) ; then, supposing afterwards the rest of the fluid in the exterior shell to be considered on JL.'s surface, the portion so condensed, if we neglect quantities of the order 8a, compared with those retained, will be
O
n 2
/
^
2
n " " ~* 5
155
THE EQUILIBRIUM OF FLUIDS. and
A's
since, in the transfer of the fluid to
surface, its particles
move over spaces of the order Sa only, the alteration which will will evidently be of the order thence be produced in
V
and consequently the value of
V will become
k being a quantity which remains
when 8a
finite
vanishes.
In establishing the preceding results, n has been supposed greater than 2, but p the density of the fluid within B and the quantity of it condensed on A's surface being still determined by the same formulae, the foregoing value of ought to hold good in virtue of the generality of analysis whatever n may be, and
V
therefore small,
when n
we
shall
is
a positive quantity and Ba
is
exceedingly
have without sensible errors
Conceiving now P' to represent the density of the fluid con2 densed on a surface, 4?ra P' will be the total quantity thereon
A
contained,
1
which being equated
to the value before given, there
results
and hence we immediately deduce +n
2"
Q represents the total quantity of redundant fluid is in the entire sphere A, the quantity contained in Moreover as
B
THE LAWS OF
156
If
now when n
is
supposed
thesis similar to Dufay's,
less
than
2,
we adopt an hypo-
and conceive that the quantities
of fluid
A
are exceedingly of opposite denominations in the interior of in this is a natural when then after having state, body great
introduced the quantity Q of redundant fluid, we may always by means of the expression just given, determine the value of Ba t
so that the
whole of the
fluid of contrary
contained in the inner sphere of it being determined by the
name
to
,
may
be
the
density in every part of the equations (12). If afterwards the whole of the fluid of the same name as Q is I?,
first
condensed upon A's surface, the value of V in the interior of as before determined will evidently be constant, provided we
B
n
Hence all neglect indefinitely small quantities of the order Sa*. will be in equilibrium, and as the shell the fluid contained in
B
A
B
and is entirely included between the two concentric spheres void of fluid, it follows that the whole system must be in equilibrium.
From what
has preceded,
we
see that the first of the formulas
(12) which served to give the density p within the sphere A when n is greater than 2, is still sensibly correct when n represents any positive quantity less than 2, provided we do not
immediate vicinity of A's surface. But as the is only approximative, and supposes the foregoing of the two fluids which originally neutralized each quantities other to be exceedingly great, we shall in the following article extend
it
to the
solution
endeavour to exhibit a rigorous solution of the problem, in case n > 2, which will be totally independent of this supposition.
Let us here in the first place conceive a spherical surface 8. whose radius is a, covered with fluid of the uniform density P', and suppose it is required to determine the value of the density p in the interior of a concentric conducting sphere, the radius of which is taken for the unit of space, so that the fluid therein
THE EQUILIBRIUM OF FLUIDS. contained,
may
be
in.
157
equilibrium in virtue of the joint action of itself, and on the exterior spherical
that contained in the sphere surface.
it
surface,
Liv.
V
now
If
ii.
is
represents the value of clear
due to the exterior Gel.
No. 12) that
_ __ ~)ff'^-(3-n)r dcr
V
from what Laplace has shown (Mec.
.
[(a
r)
>>
being an element of this surface, and g being the distance
of this element from the point
p
to
V
which
supposed to
is
belong. If afterwards fluid
we
conceive that the function
within the sphere
article, that in
consequence of the equilibrium
V'+ V=
V is
due to the
easy to prove as in the last
itself, it is
we must have
const.
V
and consequently V is of the form Y therefore by employing the method before explained (Art. 4), we get
But
(0)
,
where, as in the present case, stant quantities,
replaced
they have
by J?
Hitherto the exponent
by making
when
i
/3
=-
,
B yS
(0)
j^'
for
,
f^\ f^\
etc.
are all con-
the sake of simplicity been &c.
has remained quite arbitrary, but
the formula (11) Art.
4,
will
become
= 0,
2.3.4 - -n '71-2
sin
I
n-2.n-l...n+2t'-3
THE LAWS OF
158
Giving now taking the
sum
to
the successive values
t
0, 1, 2, 3,
and
&c.
of the functions thence resulting, there arises
V= F = F + V + F + F + etc. = 8. V (0)
(0)
(0)
(0)
3
2
n-2.n-l...n+2t'-3
n,
^
2
where the sign
8 is
referred to the variable
equation obtain
sm(
'n2
const,
*
r
to
t'.
V
F and their values and expanding the function
Again, by substituting for
V+ V
2
arid
t
y 4,
. .
.2t-2t'+2-n
which by equating separately the
V
we
n-2.n-l...
of the various,
coefficients
powers of the indeterminate quantity
in the
and reducing, gives
r,
generally
w 4
2
.
~2 Then by assigning
n
n ...2s
77.
2s
'
to
its
successive values
1, 2, 3,
results for the determination of the quantities
Q
the following system of equations,
&c ........
&c.
.
. .
&c.
&c., there
B B^ B^
.
,
&c.,
159
THE EQUILIBRIUM OF FLUIDS. evident from the form of these equations, that satisfy the whole system by making
But
it is
we determine
provided
B
Q
we may
by
2-n
2-n.^-n
_2
'
2
2
.
4
n-2
Hence
as in the present case,
/3
=
,
Si
we immediately
de-
duce the successive values
and
=l-
In the value of p just exhibited, the radius of the sphere is taken as the unit of space, but the same formula may easily be adapted to any other unit by writing
and
and recollecting at the of the equation consequence
a and
r'
respectively,
const.
= V+ V
,
-
[dv.p ^-f J
before given,
sphere
,
is
in the place of
same time that
in
f I
J
9
a quantity of the dimension
1
with regard
b being the number which represents the radius of the when we employ the new unit. Hence we obtain for a whose radius is bg, acted upon by an exterior concentric
to space
sphere
-^>
9
-=
j-
:
spherical surface of
which the radius
is a,
THE LAWS OF
160
08)
......
p
=
P' being the density of the
fluid
on the exterior
surface.
If now we conceive a conducting sphere A whose radius is and determine P' so that all the fluid of one kind, viz. that which is redundant in this sphere, may be condensed on its surface, and afterwards find 6 the radius of the interior sphere B
a,
from the condition that opposite kind,
is
it
shall just contain all the fluid of the
it
evident that each of the fluids will be in
equilibrium within A, and therefore the problem originally proposed is thus accurately solved. The reason for supposing all
name
the fluid of one
to
be completely abstracted from
JB, is
that our formulae
may represent the state of permanent equifor the librium, tendency of the forces acting within the void shell included between the surfaces and B, is to abstract
A
continually the fluid of the same
name
as that on A's surface
from the sphere B.
To prove
the truth of what has just been asserted,
we
will
begin with determining the repulsion exerted by the inner sphere itself, on any point p exterior to it, and situate at the distance r centre 0. But by what Laplace has shown (Mec. CeL No. 12) the repulsion on an exterior pointy, arising from a spherical shell of which the radius is /, thickness dr and
from Liv.
its
II.
centre is at
be measured by
will
2-jrr'drp 1
7i
.
3
d_
(r
+ r')
3 ~n
n' dr'
-
(r
- rj^
r
the general term of which -when expanded in an ascending series of the powers of -
is,
.....2.3.4.5...2*+l
-
~
and the part of the required repulsion due thereto for its value before stituting p found, become
, 2,
,
r
'
r
will,
by sub-
161
THE EQUILIBRIUM OF FLUIDS. 8P'
.
-jnow remains
It
2.3.4...2S
to find the value of the definite integral herein
r'2\-i
,
contained.
But when
f
1
is )
tions are effected f
6
l
1
J
I
by known
r'*\~
/
"?)
+1
(J2
~ /2)
formulae,
expanded, and the integra-
we
?=2 2<+2
2
5'
a2
v'
B
*'=/.
''
obtain zt
2
2r^(J
-r'
2
.
25
+
3
+ w.2s
2 )
,
+ 5+w
a4
2 //n\
/1\
r
1.3.5 l+ti.3 + w.5
?^-M\
where
made
after the integrations
equal to
+w
......
have been
2s
effected,
x ought
to
be
.
The value of the integral last found being substituted in the expression immediately preceding, and the finite integral taken relative to s from 5 = to s = co gives for the repulsion of the inner sphere,
11
THE LAWS OF
162
(1X
V
2.4.6
nf?\ " I
,.3t+l+n
2s
......
_
2
X I I
'Oft
^,2\2
'
/f
}
(
v 2
since
Bin
r(i)=V*r,
and as was before observed, x
But we have evidently by means
-
.
a
of the binomial theorem,
aV V-? _ v n - 2 .n.n + 2
2.4.6 and therefore the preceding quantity becomes
If
now we make x =
rcc ,
the same quantity
may be
Having thus the value of the repulsion due sphere
B
due to the
on an exterior point />, it fluid on A's surface. But
written
to the inner
remains to determine that this last is represented
by
,_
l-n.3-n
dr
(Mec. Gel. Liv. n. No. 12.) there results
Now by
expanding
this function
163
THE EQUILIBRIUM OF FLUIDS.
The a
-
finite
"'(
last of these
form,
by
expressions remarking that
may
readily be exhibited under
'
2.4.6
'
27~
......
a2'
/4-n\
l\
-2 <0
2.
4
.
6
25
r* 'a2'
"4.
5
V
2
J
\
2
J
2
rfl
~
.
6
...
25+3
Hence, since T () = VTT? the value of the repulsion arising from J.'s surface becomes
Now by
adding the repulsion due to the inner sphere which given by the formula (16), we obtain, (since it is evidently indifferent what variable enters into a definite integral, provided each of its limits remain unchanged) is
4-7T
112
THE LAWS OF
164
upon a particle p of positive and exterior to B. We thus
for the value of the total repulsion
fluid situate
see that
by a
within the sphere
when P'
is
which
is
force
A
positive the particle p is always impelled equal to zero at B's surface, and which
Hence, if continually increases as p recedes farther from it. fluid is separated ever so little from jB's of positive any particle surface, it has no tendency to return there, but on the contrary, it
is
force
continually impelled therefrom by a regularly increasing and consequently, as was before observed, the equilibrium ;
can not be permanent until all the positive fluid has been and carried to the surface of gradually abstracted from
B
where
it is
the sphere
A
by the non-conducting conceived to be surrounded.
retained
A
is
9
medium with which
Let now q represent the total quantity of fluid in the inner sphere, then the repulsion exerted on p by this will evidently be qr"", when r is supposed infinite. Making therefore r infinite in the expression (15), and equating the value thus obtained to the one just given, there arises
-- - 2
q
=
-47rV7r.P'a 7
r
r-
\,
fft
dx xn .
,_
(1
,=p an 2
.
When the equilibrium has become permanent, q is equal to the total quantity of that kind of fluid, which we choose to consider negative, originally introduced into the and if ; sphere
A
now ^
represent the total quantity of fluid of opposite name contained within A, we shall have, for the determination of the
two unknown quantities P' and
6,
the equations
=
and 21
and hence we are enabled
to assign accurately the
manner
in
which the two fluids will distribute themselves in the interior of A q and q lt the quantities of the fluids of opposite names originally introduced into 9.
A
being supposed given.
In the two foregoing articles
we have
determined the
THE EQUILIBRIUM OF FLUIDS. manner
in
which our hypothetical
165
fluids will distribute
them-
selves in the interior of a conducting sphere A. when in equilibrium and free from all exterior actions, but the method
employed in the former is equally applicable when the sphere In fact, if we is under the influence of any exterior forces. conceive them all resolyed into three X, Y, Z, in the direction of the co-ordinates a?, #, z of a point p, and then make, as in Art.
I,
Vwe
shall have, in consequence of the equilibrium,
nx
l
l
ny
nz
l
which, multiplied by dx, dy and dz respectively, and integrated, give const.
where
=
-
j^
Xdx + Ydy + Zdz
V+f(Xdx + Ydy + Zdz)
is
always an exact
We thus see that when X, functions
V
Y,
will be so likewise,
;
differential.
Z are given rational and entire and we may thence deduce
(Art. 5)
p
where
f is
= (l-^-y'*-z'^.f(x',y',z'},
the characteristic of a rational and entire function of
the same degree as V.
The preceding method X,
is
directly applicable
Y, Z are given explicitly in functions of x, y,
of these forces,
we may
when the forces But instead
z.
conceive the density of the fluid in the
exterior bodies as given, and thence determine the state which its action will induce in the conducting sphere A. For example,
A
to be taken as in the first place suppose the radius of the unit of space, and an exterior concentric spherical surface, of
we may
which the radius is a, to be covered with fluid of the density U"d) u"d) k e i n g a function of the two polar co-ordinates 0" and CT" of any element of the spherical surface of the same kind .
as those considered
by Laplace (Mec. CeL Liv.
in.).
Then
it is
THE LAWS OP
166
easy to perceive by what has been proved in the article last cited, that the value of the induced density will be of the form
9 TX being the polar co-ordinates of the element what U" becomes by changing 6" TX" into 0', &'. r,
',
dv,
and U'
(i)
(i}
',
Still
(Art. 4
continuing
and
5)
we
get in the present case
and by expanding )
=B +
/
Q
fP = B U",
Hence,
we have
f(r'*),
2
/(r'
methods before explained,
follow the
to
2
/
+
4
+
and
t
4-n.6-n
_
2.4.6
sin
n-2.n ...... n + 2^-4 2.4
2*
-2*'
n-l.n+1
3.5
'
......
......
2^'
......
2t'+2'+l
Then, by giving to < all the values 1, 2, 3, &c. of which it is susceptible, and taking the sum of all the resulting quantities,
we
shall have, since in the present case
F
single term
V reduces
itself to
the
(i)
,
4-n.6-n sm
I
'*-2 -
TT
~4 2.4 ...... the sign If
S
n-l.n+1 3
2t'
.
5
belonging to the unaccented
now
V represents
to the fluid
......
letter
2*
+ 2^+1
t.
F
and due the function analogous to surface, we shall obtain by what
on the spherical has been proved (Art. 3)
v>= UK> (t)
_
2W---
8
~
A ^ (0 (r>-
representing the same function as in the article just cited.
THE EQUILIBRIUM OF FLUIDS. Moreover,
it is
167
evident from the equation (10) Art.
4,
that
l-n 2
>
^..
3.5...
...2z-l
1.3
2 i+2^
2.4...
and consequently,
n"
K the
n
+*
*
now
for
V and
const.
const.
t2 '
*Q'
of equilibrium,
we immediately
'
extending from
finite integrals
Substituting
2
*
t'
=
to
t'co.
V their values
in the equation
= F'+F.
(20),
obtain
n-2.n... 2
yi-2.ro... 2
ft
+ 2'-4
4- n
2^~
2
.4...
the constant on the
except
when
i
2'
.4..
left side
.
.
6
- n ... 2<-2$' + 2-ro' 4
...2*-2
f
of this equation being equal to zero,
= 0.
By equating separately the coefficients of the various powers of the indeterminate quantity r, we get the following system of equations
:
-2 2 sin
,
^
,.
^ V-^ a
,
.-->=*, +
+
+ &c.
THE LAWS OF
168
2 sin
22.4 n
4 TT
^
.
4
w 6 .
2 sin
2.4
2
But
it
make the
is
&c
&c
&c
evident from the form of these equations, that
generally
first is,
-Z?,
if
we
= +1
i*JBt , they will all be satisfied provided this means the first equation becomes
and as by
S
1
- a*
= Et-
a3
-
there arises
2 sin 7T
Hence
am
2
(a
- 1)
2
2
(a
-/
and the required value of p becomes
But whatever the density P on the inducing spherical surface may be, we can always expand it in a series of the form
THE EQUILIBRIUM OF FLUIDS.
169
and the corresponding value of p by what precedes
will
be
2sm >
7T
+ &c.
x ZT
U'
(0)
,
U' m &c. being what
(l
\
,
0", or" into
by changing element dv.
But, since
0',
-cr',
we have
J7" (0 >,
t^
1 *,
tn tn/.i;
Z7" w , &c.
become
the polar co-ordinates of the
generally
sn (Mec. Gel. Liv. in.) the preceding expression becomes
sm[_-7r] /~2
i\
2
0"=0
the integrals being taken trom if
OT
to 0"=7r,
and from
-sr"
to
= n2?T. In order
to find the value of the finite integral entering into
the preceding formula, let
two elements series of the
2
'a
{cos 0' cos 0"
an ascending
+ sin & sin 0"cos (r f
r") j
+ 1"
*
)
ai
>
Liv. in.).
If
in -g
r powers of we shall obtain
- 2ar
V
represent the distance between the
dv; then by expanding
d
r/ o
R
now we
Hence we immediately deduce
substitute this in the value of p before given, 7
afterwards write -, and
2
and
'2
^3
in the place of their equivalents,
THE LAWS OF
170
we
shall obtain f
n
2
Bin
^ ~1 2
H
2^
2 )
(
1
~ /2
* )
j'^
3
>
the integral relative to dor being extended over the whole spherical surface.
Lastly, if p l represents the density of the reducing fluid disseminated over the space exterior to A, it is clear that we shall
by changing P into p^da in the and then integrating the whole relative preceding expression, to a. Thus, get the corresponding value of p
But do-da = dv l exterior space,
where the
dvl being an element of the volume of the and therefore we ultimately get ;
last integral is
supposed to extend over
and R the two elements dv and dv^ exterior to the sphere
It is easy to perceive
(Art. 7),
that
we may add
all
the space
to represent the distance
between
from what has before been shown to
any of the preceding values of
p,
a term of the form
h being an arbitrary constant quantity:
for it is clear
from the
which such an addition could produce would be to change the value of the constant on the left side of the general equation of equilibrium and as this article just cited, that the
only alteration
;
constant
arbitrary, it is evident that the equilibrium will not be at all affected by the change in question. Moreover, it may is
be observed, that in general the additive term
is necessary to enable us to assign the proper value of p, when Q, the quantity of redundant fluid originally introduced into the sphere, is given.
THE EQUILIBRIUM OF FLUIDS.
171
In the foregoing expressions the radius of the sphere has been taken as the unit of space, but it is very easy thence to deduce formulae adapted to any other unit, by recollecting that
^, j~^ and
,
yr^j,
are quantities of the dimensions 0,
1,
for if b rerespectively with regard to space: when we employ any other unit presents the sphere's radius, M M //7? we shall only have to write, 7 -r , -r- , -W and r in the place b o o o b
and 3
1
Ti
/-
fit
,
r, B, dv l and a, and afterwards to multiply the resulting expressions by such powers of Z>, as will reduce each of them to
of
r,
their proper dimensions.
we
here take the formula (22) of the present article as an example, there will result, If
sinpjz!^
^
2
_
s
value of the density which would be induced in a sphere A, whose radius is b by the action of any exterior bodies whatfor the
}
ever.
When n>2, the value of p or of the density of the free fluid here given offers no difficulties, but if n<2, we shall not be able strictly to realize it, for reasons before assigned (Art. 6 and 7). If however n is positive, and we adopt the hypothesis of
two
fluids,
supposing that the quantities of each contained by
bodies in a natural state are exceedingly great, we shall easily perceive by proceeding as in the last of the articles here cited, that the density given by the formula (23) will be sensibly correct except in the immediate vicinity of A's surface, provided we extend it to the surface of a sphere whose radius is b $b only,
and afterwards conceive the exterior
shell
entirely de-
prived of fluid the surface of the conducting sphere itself having such a quantity condensed upon it, that its density may every :
where be represented by
p= r>,
THE LAWS OF
172
Application of the general Methods Planes, &c.
circular conducting
to
10. Methods in every way similar to those which have been used for a sphere, are equally applicable to a circular plane, as we shall immediately proceed to show, by endeavouring in the first place to determine the value of V when the density of the fluid
on such a plane
/ being the
of the form
is
characteristic of a rational
and
entire function of the
degree s ; #', y the rectangular co-ordinates of any element da of the plane's surface, and /, & the corresponding polar coordinates.
Then we
shall readily obtain the formula n
1
a
>
where rt 6 are the polar co-ordinates of p, and the integrals are to be taken from 0' = to ff = 2-Tr, and from r to r = 1 ;
the radius of the circular plane being for greater simplicity considered as the unit of distance.
Since the function f(x, y'} is rational and entire of the deSj we may always reduce it to the form
gree
/(', y)
= AM + A
+ the coefficients
(l)
Aw A
cos
& -f
J3 (1) sin
(l
,
&+A
B
A *\ (
\
(z}
cos 2ff
sin 20'
+B
a
4
cos
sin BO'
&c. 75 (1) , J5 (2) ,
tions of r only of a degree not exceeding
'*+
+A
(3)
5,
+
3(9'
+
(24),
(3)
&c. being func, and such that
.Z?
+ &c.; A = ( +
3
(1)
(1)
4
2
4
(2)
(
l
8
.
We will now consider more particularly the part of V due any of the terms in / as A cos iff for example. The value (i)
this part will evidently
be
;
2
to
of
THE EQUILIBRIUM OF FLUIDS.
173
the limits of the integrals being the same as before. make & + <, there will result d& = d$ and
But
if
we
9
cos iO'= cos iO cos
sin id sin /<,
i(f>
and hence the double integral here given by observing that the term multiplied sin fy vanishes when the integration relative to (f>
is effected,
cos
If
A
(i]
t
n
Ad) (l}
i0f*A
now we
term a
becomes
write
.
j
'
/
V
(i}
t
-r
2V
cos
d> cos
'2 2
for that portion of
in the coefficient
r
.
'
r dr (I
tflV"** 1 dr'
1
A
(i)
we
V which
is
due to the
have
shall
-/
But by well known methods we readily get d(> cos i(>
r
- 2rr' cos + /
J o
j
a-n-i^oo
2
"
<
~....2
.
4
......
when / > r, and when r' < r, correct, provided we change r
2f
2
.
4
the same expression will into r
and
still
be
reciprocally.
V
we shall readily This value being substituted in that of have by following the processes before explained, (Art. 1 and 2), (i}
t
3
/2/8+5+2t-2A
THE LAWS OF
174
ryn-l.n+l...n 2
.
4
+ 2t'-3
n- 1 .n + 1
...
2.4
n + 2t'+ 2*'- 3
...
2i+2t'
2t'
...
3-n.5-n... the sign of integration
2
l
+ 2t-2t'-n
belonging to the variable
t'.
Having thus the part of V due to the term a cos iff in the expansion of /(a/, y ) it is clear that we may thence deduce the sin iff by simply changing part due to the analogous term b a M cos id into b M sin and we shall have the (i)
t
r
(i}
t
due to
all
V itself,
by taking
sum
the
of the various parts
the different terms which enter into the complete ex-
pansion of /(a?', If
consequently
i&,
t
t
total value of
y').
now we make
/3
=
- and recollect that
sm
-_
the foregoing expression will undergo simplifications analogous to those before noticed (Art. 5). Thus we shall obtain
sm
(-33
2.4... or
by
...
2^+2*'
3-^. 5 -n...l +2t- 2t'-n
2.4
...
writing for abridgment
,^_ 2
.
4
2.4
there will result this particular value of
and afterwards by making
/3
...
THE EQUILIBRIUM OF FLUIDS.
we
shall
175
have ;
cos id into
x
. .
sn
2
4
.
n.5
3
n.l
n
+ &c.,
+ &c
&c
Conceiving in the next place that
V is
a given rational and
entire function of x, y, the rectangular co-ordinates of jp 9 shall have since x r cos 0, y r sin 6,
we
=
V=
(7
(0)
+ <7
(1)
+E
(l)
cos
+
sin
+E
(7
(2)
cos 20
+
20
+E
sin
of which expansion any coefficient as still farther developed in the form
{3)
(S)
(7
+ &c.
cos
3(9
sin
30 + &c ....... (25),
(0
for
example,
may
be
Z-; 2)
Now
it is
clear that the
term
(7
(i)
cos iO in the developement
V which we have designated and hence by equating these two forms of the same
(25) corresponds to that part of
by
F
(i)
,
quantity,
we
get
which by substituting
for
V
(i)
and
C
(i)
their values before ex-
THE LAWS OP
176 hibited, tity
and comparing
n
3 t
(t) "
2
= 1>a
&c.= of
powers of the indeterminate quan-
r gives
l= C (i)
ci (i)
like
3
i
.........
.
a*
(i) '
2
.
4
a
3n.5n.7n a 2
.
4
6
.
(i
*
3-n.5-n
2.4
fl
2
n
n.5
3 (i) "
~^
(i), i
^
a
&c .......... &c.
which system the general type
is
the symbols of operation being here separated from those of quantity, and e being used in its ordinary acceptation with reference to the lower index w, so that '". <*!?
we
shall
have generally
= <.
w general equation between ajf and c being resolved, evidently gives by expanding the binomial and writing in the ! place of ec?, e c', e'c" , &c. their values c&,, c.s , <$,, &c.
The
1
Having thus the value of oj, we thence immediately deduce the value of A and this quantity being known, the first line of the expansion (25) evidently becomes known. 1'
(i}
In like manner when we suppose that the quantity expanded in a series of the form
;
EP 9
is
2).r'+&c.}
THE EQUILIBRIUM OF FLUIDS.
we
177
shall readily deduce
and
b
being thus given,
'B
w and consequently the second
line
of the expansion (25) are also given.
From what
has preceded,
and
it
is
when V is given whatever of x and y>
clear that
entire function
equal to any rational the value off(x, y'} entering into the expression
will immediately be determined
by means
of the most simple
formulae.
The preceding infinite, or
being quite independent of the degree y) will be equally applicable when s is
results
s of the function f(x',
wherever this function can be expanded in a series x y, and the various products of these
of the entire powers of
,
powers.
We will
now endeavour
one fluid will distribute
A
when
acted upon
by
to
itself
determine the manner in which
on the circular conducting plane
fluid distributed in
any way
in
its
own
plane.
For
this purpose, let us in the first place conceive a
quan-
=
=
in a point P, where r a and 6 0, tity q of fluid concentrated Then to act upon a conducting plate whose radius is unity. be this fluid will to due the value of evidently
V
V' 2
(a
2ar cos 6
+ ra
* )
and consequently the equation of equilibrium analogous one marked (20) Art. -10, will be const.
V being
due
=-
to the fluid
2
^+ F
to the
(27)
on the conducting plate only. 12
;
ON THE LAWS OF
178 If tion,
now we expand
and then compare it we shall have generally
article,
= 0,
V
deduced from this equathe formulae with (25) of the present
the value of
E
(i}
0,
and
which case we must take only half the have the corquantity furnished by Hence whatever u may be, rect value of <7 except
when
i
in
this expression in order to
(0)
.
and
}
c
=7T
= being excepted, for in this case the particular value to the preceding remark
we
have agreeably
n-1
., and then the only remaining exception is that due to the conon the left side of the equation (27). But it will be more simple to avoid considering this last exception here, and to afterwards add to the final result the term which arises from the constant quantity thus neglected. stant quantity
The for
equation (26) of the present article gives by substituting value just found,
c[? its
2 sin
n-3 .n-1
n-3.n-l.n-l
2 sin
8-M
THE EQUILIBRIUM OF FLUIDS.
179
and consequently,
A* =
K
(i)
+
2
fi
m(fn-l 2sm
,
,2
4
7T
the particular value result
^
(0)
being one half only of what would
from making t'=0 in this general formula.
= 0, and therefore the expanevidently gives jE' sion of/(o;', y] before given becomes But eS (
f (V,
=
(i)
=A +A (0)
y')
/TO
1
{1]
cos
^+ A
\
2sm^-7r) \
cos
2(9'
+^
(3)
.
/
2
i\
2
/_2
+ &c.
'2\-l
+^ by summing the
3(9'
^
7
or
cos
series included
cos 2^'
+ &c.i
between the braces,
_ gar' cos (9'+ r'
2
n-l
R
being the distance between P, the point in which the quanand that to which the density p is tity of fluid q is concentrated,
supposed to belong.
122
ON THE LAWS OF
180
Having thus the value of / (x, y) we thence deduce
n-1
.
sm
The
value of p here given being expressed in quantities of the axis from which perfectly independent of the situation the angle & is measured, is evidently applicable when the point is not situated upon this axis, and in order to have the com-
P
to add the term plete value of p, it will now only be requisite left side of the the due to the arbitrary constant quantity on
that equation (26), and as it is clear from what has preceded, the term in question is of the form --3
const,
we
shall therefore
x
(1
.
The
*
z
)
,
have generally, wherever
= (1 - r'V
n
Pmay
be placed,
1
8
-* p
r
( .
const.
transition from this particular case to the
more general
one, originally proposed is almost immediate : for if p represents the density of the inducing fluid on any element dcr^ of the plane
coinciding with that of the plate, p l d(r i will be the quantity of fluid contained in this element, and the density induced thereby will be had from the last formula, by changing q into p^cr^.
we integrate the expression thus obtained, and extend the integral over all the fluid acting on the plate, we shall have for the required value of p If then
7T
ft
being the distance of the element dv^ from the point to which
p belongs, and a the distance between da^ and the center of the conducting plate. Hitherto the radius of the circular plate has been taken as if we employ any other unit, and sup-
the unit of distance, but
THE EQUILIBRIUM OF FLUIDS. pose that b
da^ and
R
ft-
>
r
>
-TT and j- in the place of is
a,
we /,
a quantity of the
with regard to space, by so doing the resulting is
- ,/
-
--7T
Sinl
n-3
=
to write r
respectively, recollecting that
dimension value of p
the measure of the same radius, in this case
is
have
shall only
181
( .
const.
= supposing w
--
-
.....
(28).
the preceding investigation will be applicable to the electric fluid, and the value of the density induced upon an infinitely thin conducting plate by the action of a quan-
By
2,
tity of this fluid, distributed in
any way
the plate itself will be immediately given. the foregoing value of p becomes
p= If
at will in the plane of
In
fact,
when n = 2,
w=7*\
we suppose
all extraneous action, we in the preceding formula;
the plate free from
shall simply have to
make
/o t
=
and thus ,(29).
Biot (Traiti de Physique, Tom.
II.
p. 277),
has related the
some experiments made by Coulomb on the distribution of the electric fluid when in equilibrium upon a plate of copper 10 inches in diameter, but of which the thickness is not If we conceive this thickness to be very small comspecified. with the diameter of the plate, which was undoubtedly pared
results of
the case, the formula just found ought to be applicable to it, provided we except those parts of the plate which are in the immediate vicinity of its exterior edge. As the comparison of
deduced from the received theory of with of those the experiments of so accurate an obelectricity as server Coulomb must always be interesting, we will here give
any
results mathematically
ON THE LAWS OP
132
a table of the values of the density at different points on the surface of the plate, calculated by means of the formula (29), values found from experiment. together with the corresponding
We thus
see that the differences
observed densities are trifling
between the calculated and
and moreover, that the observed
;
are all something smaller than the calculated ones, which it is evident ought to be the case, since the latter have been deter-
mined by considering the thickness of the plate as infinitely small, and consequently they will be somewhat greater than when this thickness is a finite quantity, as it necessarily was in Coulomb's experiments. has already been remarked that the method given in the
It
second
article
axes are a,
applicable to
is
In
5, c.
co-ordinates of a point
element dv of
we
its
any
within
p
ellipsoid whatever,
we suppose it,
whose
that x, y, z are the #', y ', z those of any
and
volume, and afterwards make
x=a
.
cos
0,
x=a
.
cos
ff ,
y =
b
by
sin
.
y =b
shall readily obtain
F= abcfp
if
fact,
.
sin
6 cos
& cos
tn-',
z
c
z
.
c
.
sin 6 sin sin
w,
& sin <*/,
substitution,
z
.
OT,
r dr'de'd
- 2/m-' + vr'^
;
the limits of the integrals being the same as before (Art. 2), and
X = a2 cos
=a /i v =a
2
2
2
+
2
sin
2 tf
cos *?
+ c* sin & sin ^
&+ b* sin 6 sin ^cos w cos r'+ c + tf sin 0' 2 cos + c sin 0"* sin
2
cos 6 cos cos &*
2
2
-cr'
2
,
sin 6 sin 0'sin CT sin OT', '2
.
THE EQUILIBRIUM OP FLUIDS. Under the present form
it is
183
clear the determination of
V
what has been shown (Art. 2). I shall not therefore insist upon it here more particularly, as it is my intention in a future paper to give a general and purely method of analytical finding the value of F, whether p is situated can
offer
no
difficulties
after
within the ellipsoid or not. for the particular value
the series
U '+
single term
J7 ',
Z72 '+
I shall therefore only observe, that
U + &c.
and we
'
4
(Art. 2) will reduce itself to the
shall ultimately get
2 sin
Hence it follows that evidently a constant quantity. the expression (30) gives the value of p when the fluid is in equilibrium within the ellipsoid, and free from all extraneous which
is
action.
Moreover, this value
is subject,
fications similar to those of the (Art. 7).
when n <
2, to
modi-
analogous value for the sphere
ON THE DETERMINATION OF THE
EXTERIOR AND INTERIOR
ATTRACTIONS OF ELLIPSOIDS OF
VARIABLE DENSITIES.
*
From
the Transactions of the Cambridge Philosophical Society, 1835.
[Read
May
6,
1833.]
ON THE DETERMINATION OF THE EXTERIOR AND ATTRACTIONS VARIABLE DENSITIES.
INTERIOR
THE
OF
ELLIPSOIDS
OF
determination of the attractions of ellipsoids, even on
the hypothesis of a uniform density, has, on account of the utility and difficulty of the problem, engaged the attention of
the greatest mathematicians.
Its solution, first
attempted by
Newton, has been improved by the successive labours of Maclaurin, d'Alembert, Lagrange, Legendre, Laplace, and Ivory. Before presenting a new solution of such a problem, it will naturally be expected that I should explain in some degree the nature of the method to be employed for that end, in the follow-
ing paper; and this explanation will be the more requisite, because, from a fear of encroaching too much upon the Society's time, some very comprehensive analytical theorems have been in the first instance given in all their generality. It is well known, that when the attracted point
p
is
situated
within the ellipsoid, the solution of the problem is comparatively easy, but that from a breach of the law of continuity in the values of the attractions
when p
passes from the interior of the
ellipsoid into the exterior space, the functions by which these attractions are given in the former case will not apply to the
As however this violation of the law of continuity always be avoided by simply adding a positive quantity, uz for instance, to that under the radical signs in the original
latter.
may
seemed probable that some advantage might thus be and the attractions in both cases, deduced from one common formula which would only require the auxiliary variable u to become evanescent in the final result. The principal which however arises from the introduction of the new advantage integrals,
obtained,
it
ON THE DETERMINATION OF THE ATTRACTIONS
188
variable u, depends on the property which a certain function V* then possesses of satisfying a partial differential equation, when-
ever the law of the attraction
is
inversely as any power n of the
For by a proper application
distance.
of this equation
we may
the difficulty usually presented by the integrations, same time find the required attractions when the density p is expressed by the product of two factors, one of which is a simple algebraic quantity, and the remaining one any
avoid
and
all
at the
rational
and
entire function of the rectangular co-ordinates of
the element to which p belongs.
The
original
problem being thus brought completely within is no longer confined as it were to the three In fact, p may represent a function of any
the pale of analysis, dimensions of space.
number s, of independent variables, each of which may be marked with an accent, in order to distinguish this first system from another system of s analogous and unaccented variables, to be afterwards noticed, and
V
may represent the value of a multiple integral of s dimensions, of which every element is expressed by a fraction having for numerator the continued into the elements of all the accented variables,
product of p
and
denominator a quantity containing the whole of these, with the unaccented ones also formed exactly on the model of for
V
the corresponding one in the value of belonging to the original problem. the now auxiliary variable u is introSupposing
duced, and the s integrations are effected, then will the resulting value of be a funtion of u and of the s unaccented variables to
V
*
This function in
its
original
form
is
given by
p'dx'dy'dz'
where dx'dy'dz' represents the volume of any element of the attracting body of which f> is the density and x y', z' are the rectangular co-ordinates x, y, z being 1
,
;
the co-ordinates of the attracted point p. variable
u which
is
to be
made equal
But when we introduce the
auxiliary
to zero in the final result, p'dx'dy'dz'
I,
both integrals being supposed to extend over the whole volume of the attracting body.
OF ELLIPSOIDS OF VARIABLE DENSITIES.
But
be determined.
189
after the introduction of w, the function
V
has the property of satisfying a partial differential equation of the second order, and by an application of the Calculus of Variations
it
will be proved in the sequel that the required
V may
always be obtained by merely satisfying this other simple conditions when p is equal to and certain equation, the product of two factors, one of which may be any rational
value of
and
entire function of the s accented variables, the remaining one being a simple algebraic function whose form continues unchanged, whatever that of the first factor may be.
The
chief object of the present paper is to resolve the problem in the more extended signification which we have endeavoured to explain in the preceding paragraph, and, as is by
no means unusual, the simplicity of the conclusions corresponds with the generality of the method employed in obtaining them. For when we introduce other variables connected with the original ones -by the most simple relations, the rational and entire factor in p still remains rational and entire of the same
degree, and may under its altered form be expanded in a series of a finite number of similar quantities, to each of which there corresponds a term in V, expressed by the product of two factors
new
;
the
first
being a rational and entire function of s of the and the second a function of the
variables entering into V,
remaining new variable ^, whose differential coefficient is an Moreover the first is immediately deducible algebraic quantity. from the corresponding. part
of,/)'
without calculation.
The
solution of the problem in its extended signification being thus completed, no difficulties can arise in applying it to have therefore on the present occasion particular cases.
We
given two applications only.
In the
first,
which
relates to the
attractions of ellipsoids, both the interior and exterior ones are comprised in a common formula agreeably to a preceding obser-
and the discontinuity before noticed falls upon one of the independent variables, in functions of which both these attrac-
vation,
tions are expressed
;
this variable
so long as the attracted point
becoming equal p,
when p
is
being constantly equal to zero within the ellipsoid, but
p remains
to a determinate function of the co-ordinates of
situated in the exterior space.
Instead too of seek-
ON THE DETERMINATION OF THE ATTRACTIONS
190
all its differentials have first been ing directly the value of V, of the value obtained by integration. deduced, and thence
V
This slight modification has been given to our method, both because it renders the determination of V in the case considered
more
easy,
and may likewise be usefully employed in the more
The other application is remarkgeneral one before mentioned. able both on account of the simplicity of the results to which it leads,
and of
their analogy with those obtained
by Laplace.
In fact, it would be easy to shew (Mec. Cel. Liv. ill. Chap. 2). that these last are only particular cases of the more general ones
now under
contained in the article
notice.
The
general solution of the partial differential equation of second the order, deducible from the seventh and three following
and in which the principal variable
articles of this paper,
V is
s + 1 independent variables, is capable of being applied with advantage to various interesting physico-matheIndeed the law of the distribution of heat in matical enquiries.
a function of
a body of ellipsoidal figure, and that of the motion of a nonelastic fluid over a solid obstacle of similar form, may be thence almost immediately deduced; but the length of our paper entirely precludes
any thing more than an
allusion to these appli-
cations on the present occasion.
The
object of the present paper will be to exhibit certain formulae, from which may be deduced as a analytical general the values of the attractions exerted by case very particular 1.
ellipsoids
upon any
exterior or interior point, supposing their
densities to be represented by functions of great generality. Let us therefore begin with considering p as a function of
the s independent variables
and
let
us afterwards form the function '
'
'
'
n P
the sign / serving to indicate s integrations relative to the variables a?/, a?2', #8', ... xt and similar to the double and triple ones '
t
OF ELLIPSOIDS OF VARIABLE DENSITIES.
191
employed in the solution of geometrical and mechanical problems.
Then
easy to perceive that the function
it is
V will
satisfy the
partial differential equation .
~~^
+
+
^
n-sdV"
seeing that in consequence of the denominator of the expression elements satisfies for Fto the equation (2). (1), every one of its To give an example of the manner in which the multiple integral is to be taken, we may conceive it to comprise all the real values both positive and negative of the variables oj/ajj',
...
which
#/,
the symbol <, as
satisfy the condition
what
follows, not excluding
difficulties
usually attendant on
the case also in
is
equality.
2.
In order to avoid the
integrations like those of the formula (1), it will here be convenient to notice two or three very simple properties of the function V.
In the
first
place, then,
it
clear that the denominator of
is
(1) may always be expanded in an ascending series of the entire powers of the increments of the variables x^ a;2 ,...aj8
the formula
,
u,
and
unless
their various products
by means
of Taylor's
we have simultaneously = #/, #2 = tf2 ...... X8 = x8 and w = #!
Theorem,
'
',
and therefore
V may
form, unless the s satisfied for
^2
series of like
+1
V
possesses the property in question, except the two conditions T-
2
~2
2
+ a + u?-' + - + u? i<1 ?* ?* cfcj
0;
equations immediately preceding are all one at least of the elements of F. It is thus evident
that the function
only when
always be expanded in a
2
8
and
u=0 ............
8
are satisfied simultaneously, considering as
we
shall in
what
ON THE DETERMINATION OF THE ATTRACTIONS
192
follows the limits of the multiple integral (1) to be determined by the condition (a)*.
In
like
manner
it is
clear that
when
V
in powers of u will contain none but the the expansion of even powers of this variable.
Again,
is
it
V
quite evident from the form of the function of the -s + 1 independent variables therein
when any one
that
contained becomes infinite, this function will vanish of
The
3.
itself.
V
three foregoing properties of combined with the will furnish some useful results. In fact, let us
equation (2) consider the quantity
where the multiple integral comprises
all
the real values whether
positive or negative of o^, &,,...# with all the real values of u which satisfy the condition
ax
a2
,
,
. . .
ag and h being positive constant quantities
we may have
that
and
;
positive
and such
generally
ar
>
a/.
In this case the multiple integral (5) will have two extreme limits, viz. one in which the conditions
xz -*$
*
xz
+ -** +
.
+
X* -*i
u*
+ j- = l a
and u = a positive quantity...
(7)
property does not explicitly appear in what follows, in order to place the application of the method of integration by parts, in Nos. 3, 4, and 5, beyond the reach of objection. In fact, when V possesses this property, the theorems demonstrated in these Nos. are certainly correct but they are not necessarily so for every form of the function V, as will
but
The necessity of this must be understood
first
it
:
be evident from what has been shewn in the third article of Application of Mathematical Analysis [See pp. 2327.]
to the
my
Theories of Electricity
Essay On
the
and Magnetism.
OF ELLIPSOIDS OF VARIABLE DENSITIES. are satisfied
;
193
and another defined by ^2 al
+ + ^%+... a z
as2
<1
and u
= 0.
Moreover, for greater distinctness, we shall mark the quanbelonging to the former with two accents, and those be-
tities
longing to the latter with one only.
Let us now suppose that V" is completely given, and likewise V[ or that portion of in which the condition (3) is as quite satisfied; then if we regard F2 or the rest of
V
V
'
and afterwards endeavour
arbitrary,
a
minimum, we
to
shall get in the usual
make
the quantity
(5)
way, by applying the
Calculus of Variations,
y-^-j^r du2
--
n-s dV] u
dx?
The
first line
)
(8),
iU
seeing that SF"=0 and F/ are supposed given.
r~r
du
5F/ = 0, because the
quantities
V" and
of the expression immediately preceding gives
generally ,
du which
is
identical with the equation (2)
No.
1,
and the second
line gives
dV = u' n -~'
From
(u being evanescent) ............ (9) .
the nature of the question de minima just resolved, little doubt but that the equations (2') and (9) will
there can be suffice for
the complete determination of F, where F" and F/ But as the truth of this will be of consequence we will, before proceeding farther, give a de-
are both given. in what follows,
monstration of
it
;
and the more willingly because
it is
simple
and very general. 4.
Now
since in the expression (5) u
is
always positive, 13
ON THE DETERMINATION OF THE ATTRACTIONS
194
every one of the elements of this expression will therefore be positive; and as moreover necessarily exist a function
V" and F/
F
are given, there
must
which
will render the quantity (5) follows, from the principles of the
a proper minimum. But it Calculus of Variations, that this function
F
,
whatever
it
may
must moreover
If then satisfy the equations (2') and (9). there exists any other function Vv which satisfies the last-named equations, and the given values of V" and F/, it is easy to per-
be,
ceive that the function
will do so likewise,
whatever the value of the arbitrary constant
A
A
quantity originally equal may be. Suppose therefore that to zero is augmented successively by the infinitely small increwill be ments BAj then the corresponding increment of
F
and the quantity (5) will remain constantly equal to its minimum however great A may become, seeing that by what precedes the variation of this quantity must be equal to zero what-
value,
ever the variation of are all satisfied. tion
F
V
all, we might give to the partial difhowever great, by augmenting the values any thus cause the quantity (5) to exceed and sufficiently,
satisfying
t
the foregoing conditions there exists another func-
Fmay be, provided
If then, besides
them
ferentials of F,
quantity
A
finite positive one,
any Hence no such value
contrary to what has just been proved.
as
F
t
exists.
We
thus see that when F" and F/ are both given, there is one and only one way of satisfying simultaneously the partial differential equation (2), 5.
it
Again,
the whole of
F
2';
is
and the condition
it
has before been observed (No. 2) that the formula (1), it may always be ex-
panded in a
series of the
Hence the
right side of the equation (9)
1
;
.
clear that the condition (4) is satisfied for
and
when Fis determined by
order w" '*1
(9).
form
and u being evanescent,
is
a quantity of the
this equation will then
OF ELLIPSOIDS OF VARIABLE DENSITIES.
we
evidently be satisfied, provided follows, that
n
s
+1
we
suppose, as
195
shall in
what
is positive.
If now' we could by any means determine the values of V" and F/ belonging to the expression (1), the value of V would be had without integration by simply satisfying (2') and (9), as is evident from what precedes. But by supposing all the constant quantities alt 8 as ...... a, and h infinite, it is clear that we shall have ,
= and then we have only
V",
to find F/,
and thence deduce the gene-
ral value of F.
For
6.
this purpose let us consider the quantity
dV dU d_v_du 7 ~~J dx 8 axa
dvdm T^
7 7 du du
**
r
I
Av I
)
the limits of the multiple integral being the same as those of the expression (5), and being a function of xl9 a?2 ...... xs and
U
the condition
u, satisfying
,
= U" when
al9
2
,
......
aa and h
are infinite.
But the method
of integration
by
parts reduces the quan-
tity (10) to
xt ...... dxa
since
= F"
tity (10)
and as we have likewise under the form may ;
=
U", the same quan-
also be put
n-sdY
^
132
ON THE DETERMINATION OF THE ATTRACTIONS
196
U
like V also satisfies the equation each of the expressions (11) and (12) will be reduced to its upper line, and we shall get by equating these two forms of the
Supposing therefore that
(2'),
same quantity
:
was before ex-
the quantities bearing an accent belonging, as plained, to one of the extreme limits.
V
Because
satisfies
diately preceding
If
now we
the condition
(9),
imme-
the equation
be written
may
give to the general function
U
the particular
value l-n
which is admissible, since it satisfies for V to the equation and gives U" = 0, the last formula will become
(2),
du
dx,dx,...dx8
.(l-n)u
>n
-"V n+i
in
which expression
u'
\
L{>)>
must be regarded as an evanescent
posi-
tive quantity.
In order now to
effect the integrations indicated in the second of this equation, let us make
member
" = up sin O cos 6 = up cos 6 a? a? 2 " = up sin 0j sin cos &c. o? o?
05,
05/'
1 ;
3
until
we
x
.
w'
x" t_l
&
3
2
two
,
last, viz.,
= up sin O
l
up
Z
l
3
arrive at the #,_!
2
sin
sin 6\ sin
2
2
. . .
sin 0,_ 2 cos 0,_ 1}
. . .
sin 0,_2 sin
being, as before, a vanishing quantity.
^
;
OF ELLIPSOIDS OF VARIABLE DENSITIES.
Then by
the ordinary formulas for the transformation of
multiple integrals
we
get sin
6'*
sin
df...
sin
and the second number of the equation become dp de i d02
197
d6s_, p'- 1
...
2
sin 6>/- sin ,
Of
. . .
0\^dpd0^dOz ...dO'a-V t
(13)
by
sin 0,
2
.
substitution will
-*)
(1
V
., 4 . t m
But ever x^ over n
u
since
x2 ,...x8
+1
is
we shall have p " from x", x^...xa
evanescent,
differ sensibly
4)
whenand as more-
infinite, ;
positive, it is easy to perceive that we may all the neglect parts of the last integral for which these difs
is
Hence
ferences are sensible. stant value
0,_j
V
'
in
V may be
replaced with the con-
which we have generally
Again, because the integrals in (14) ought to be taken from = to 0,_j = 2?r, and afterwards from 6r = to 6r = TT, what-
ever whole number less than 5 easily obtain
/sin
by means
1
of the well
Of sin Of sin Of
......
be represented by
may
known
sin 0.
2
dO,
function
dO%
...
dd,
Gamma 1
=
and as by the aid of the same function we readily get
r
-
(*}
rL
the integral (14) will in consequence become
r and thus the equation
(13) will take the
form
r, :
we
198
ON THE DETERMINATION OF THE ATTRACTIONS j
L
,
In
ax,
- a,")" + fo -
V is
this equation
. . .
i dx 8u
'n *
JV 1
=-*-
du
o +-+
i
1
(*
- O* +
'2)
2
supposed to be such a function of xv
...xs and w, that the equation (2) and condition (9) are both is the particular value of Moreover V" = 0, and satisfied.
xz
F
for
V
'
which
x
t
= x"
;
#2
Let us now make,
and afterwards change
= a? " 2
;.
..... a?g
=
a;,",
and u
= 0.
for abridgment,
a?
into a/,
and x" into x in the expression
immediately preceding, there will then result
.'2)
P' being what
P becomes
2
by changing generally xr into a?/, P' indicating, as before, that
unit attached to the foot of
the the
multiple integral comprises only the values admitted by the condition (a), and V' being what becomes when we make
V
w
= 0.
The equation just given supposes u' evanescent but if we were to replace u with the general value u in the first member, and make a corresponding change in the second by replacing with the general value F, this equation would still be correct, and we should thus have ;
V
199
OF ELLIPSOIDS OF VARIABLE DENSITIES. dx^dx,' i
(fa.'P/
- X,Y +
< - xtf +
. . .
+
(x;
- X,Y + u'F *
.
V.
(16).
() For under the present form both its members evidently satisfy the equation (2), the condition (9), and give V" = 0. Moreover, when the condition (3) is satisfied, the same members are equal
Hence by what has before been in consequence of (15). (No. 4), they are necessarily equal in general.
proved
comparing the equation (16) with the formula (1), it will evident, that whenever we can by any means obtain a value of V satisfying the foregoing conditions, we shall always be able to assign a value of p which substituted in (1) shall
By
become
reproduce this value of V. foot of P',
we
which
In
by omitting the
fact,
unit at the
only serves to indicate the limits of the integral,
readily see that the required value of p is
foregoing results being obtained, it will now be convenient to introduce other independent variables in the place 7.
The
of the original ones, such that *i
al9
2
=
i?i>
xz =
,&,*. =
u
&
,...aa being functions of h, one of the
= hv, new independent
variables, determined by
and v a function of the remaining new
variables, fx ,
f2 fs ,
,
... f,
satisfying the equation
=u + 2
1
o/, a/, a/,
equation
...
(a),
2
?!
2
+? +
+f
B
2
;
a/ being the same constant quantities as in the 1. Moreover, ad a2 ... aa will take the values
No.
,
,
ON THE DETERMINATION OF THE ATTRACTIONS
200
belonging to the extreme limit before marked with two accents,
by simply assigning The easiest way
to
h an
infinite value.
of transforming the equation (2) will be to the general one which presents itself when we apply the Calculus of Variations to the quantity (5), in order to render it a minimum. have therefore in the first place
remark, that
it is
We
and by the ordinary formula
for the transformation of multiple
integrals,
. . .
dxtdu
But the expression
=
since
1
(5) after
- 2/+1 -Hp- = ^ + #2/" substitution will
become
'
+ w, )(,/
*/
(V
-v
s,-*
Applying now the method of integration by parts to tlie variaby reduction, we get for the equivalent of
tion of this quantity, (2)
the equation
where the finite integrals are r = s + 1, and from r'= 1 to r
all
supposed taken from r
s -f 1.
=
1
to
201
OF ELLIPSOIDS OF VARIAPLE DENSITIES.
The
last
we have
provided
^
equation
.
f.
coefficient of
.
V
m y Tr ^= -^ ay
,.
coefficient of
.
in
dV in .
^r
U \
tt r
F= TT
2
,
f/ 4-
a^
f
r
n
+ <* 2, t
a/ Z
a? \
when we employ
and therefore the condition
>
f
r
^r
^
ar
-
2\
fr
,
j
a r a?
y V -^
dl;r
3loreover,
fz - K? <+l a - 6^
I/-,
.
,
,.
.
be put under the abridged form,
generally
.
coefficient of
^
d
may
(9)
the
new
a/
a -^
r'*\
a*J
variables
manner
in like
will
become
where the values of the variables fx f2 ,...fs must be such as 2 = 0, whatever h may be and as n s + 1 satisfy the equation v ,
;
is positive, it is
clear that this condition will
always be
V relative
provided the partial differentials of
to the
satisfied,
new
vari-
ables are all finite. 8.
Let us now try whether
tion (2'")
it is
possible to satisfy the equa-
by means of a function of the form 08);
H depending on the variable h only, entire function of f1? f2 ,...f
pendent of
By
we
and
being a rational and
of the degree 7, and quite inde-
h.
substituting this value of
readily get
=
V in
v-*0
(2'")
and making
........................ (18);
ON THE DETEKMINATION OF THE ATTRACTIONS
202
where, in virtue of (17) K must necessarily be a function of h only ; and as the required value of <, if it exist, must be inde-
pendent of
h,
we
have, by
making h
=
in the equation
imme-
diately preceding,
k being the value
#,
and
We shall demonstrate
when h = 0.
v'
almost immediately that every function
$ of the form (20), No. 9, which satisfies the equation (19), and which therefore is independent of h, will likewise satisfy the equation (18) ; and the corresponding value of K obtained from the latter being substituted in the ordinary differential equation (17), we shall only have to integrate this last in order to have a
proper value of V. 9.
To
satisfy the equation (19) let us ",
F being
Is
assume
,-?. ) &, &, &
8
2
............. (20)
;
the characteristic of a rational and entire function of
the degree 2y', and the most general of its kind, and f-p fa &c. designating the variables in which are affected with odd expo,
nents only
;
so that if their
number be
v
we
shall
,
have
the remaining variables having none but even exponents. Then easy to perceive, that after substitution the second member
it is
of the equation (19) will be precisely of the same form as the assumed value of <, and by equating separately to zero the coefficients of the various powers and products of ?2 ,...fg we ,
,
same number of linear algebraic equations as there are coefficients in and consequently be enabled to deshall obtain just the
,
termine the ratios of these coefficients together with the constant quantity k
In
fact,
.
by writing
* = SA mi
,
the foregoing value of
,....
<
.&.-. ...... fc*
and proceeding as above described, the ?,""&"*
......
under the form ............ (20');
coefficient of
fc*
OF ELLIPSOIDS
01'
203
VARIABLE DENSITIES.
will give the general equation
Wto
-"wi, ma,
ar '2
**\
... mr+2,
.
.
.
* * '
,
,
\
/
m>
the double finite integral comprising all the values of r and r', = r', and consequently containing when except those in which r
completely expanded s
For the terms number is
(s
1)
terms.
of the highest degree
7 and
of
which the
7+1.7+2. ........ 7+*-l_7 1.2.3 the last line of the expression (21) evidently vanishes, and thus obtain ^7" distinct linear equations between the coefficients of
we
the degree 7 in
<
and &
.
Moreover, from the form of these equations it is evident that obtain by elimination one equation in k of the degree
we may
N of which each of the N roots will give a distinct value of the 9
having one arbitrary constant for factor; the homofunction geneous M being composed of all the terms of the in But the coefficients of and Jc being highest degree, 7
M,
$M
known, we may thence easily deduce all the remaining cients in <, by means of the formula (21).
coeffi-
N
linear equations have no terms except Now, since the are factors, it follows that those of which the coefficients of
<M
if k were taken at will, the resulting values of all these coeffiIf however we obtain the values cients would be equal to zero. in of the remaining one A from terms 1 of the coefficients of the the 1 of ordinary formulas, and substitute equations, by
N
N
these in the remaining equation,
we
shall get a result of the
form
K.A = 0, where
K
is
a function of &
of the degree N.
We
shall thus
ON THE DETERMINATION OF THE ATTRACTIONS
204
have only two cases to consider First, that in which A=0, and consequently also all the other coefficients of <M together with the remaining ones in , as will be evident from the formula :
(21).
Hence, in this case
= 0:
<
Secondly, that in which '
instance,
Jc
Jc
Q
in this case all the coefficients of
we
multiples of A, and
shall
(f>
= K, will
as for
become
have
<^ being a determinate function of ft
We
N roots of
one of the
is
f2
,
,
......
fg
.
when we
consider functions of the form (20) only, the most general solution that the equation
thus see that
=V admits
is
>-^
...................... (19')
..................
or,
<
=
or,
;
a being a quantity independent of function which satisfies for
=
;
f2
,
......
,
and
(
to the equation (19').
<
any But by <
affecting both sides of the equation
we
with the symbol v>
o
get
= v.v'<-A'-v
and we shall afterwards prove the operations indicated by V' to be such, that whatever may be,
y
and
Hence, the
and as
y like
last
equation becomes
is
of the form (20),
it
follows from
what has
just been shewn, that either
= v<,
or,
v = a
a being a quantity independent of fx
fore
The first is inadmissible, since when satisfies (19'), we have
,
it
f2
,
...... (
would give
<'
= a<>,
i.e.
=
, .
7<>
a<>.
= (/>
0; there-
205
OF ELLIPSOIDS OP VAEIABLE DENSITIES.
But
independent of ft f2 ...... f8 this last equation is evidently identical with (18), since the equation (18) f2 ...... f8 merely requires that K should be independent of since a
is
,
,
,
,
Having thus proved
.
,
that every function of the form (20)
(19) will likewise satisfy (18), it will be more to the remaining coefficients of <j> from those of determine simple <(Y) by means of the last equation, than to employ the formula
which
satisfies
(21) for that purpose. Jc
therefore
Making place of K,
we
h
infinite in (18),
If
s indices
now we
r
s (s ^
taken in
1)
combinations which can be
pairs.
substitute the value of
recollect that for the
2w = 7, we
before given terms of the highest degree
(20'),
and
we have
shall readily get
.
from which
in the
get
where (22) comprises the formed of the
and writing
all
deduced, when
the remaining coefficients in
ms
...... (22),
will readily be
those of the part M are known.
10. It now remains, as was before observed, to integrate the ordinary differential equation (17) No. 8. But, by the known theory of linear equations, the integration of (17) will always
become more simple when we have a particular value satisfying it, and fortunately in the present case such a value may always be obtained from
by simply changing fr
into
-
r
Jf
fact,
if
we
represent the value thus obtained
have
dh
by
H
Q
.
/2
\/(2,ar
In
)
we
shall
ON THE DETERMINATION OF THE ATTRACTIONS
206
and by a second
differentiation
i + -**
*
rff?-^W
O
2) as before comprising
all
the
*
O
I
-L.
1
1*3
combinations of the
s
indices taken in pairs.
Hence, the quantity on the right side of the equation becomes when we make
(17),
H=E^
ar"
But
if
we
V
we have
recollect that
*
generally (24) '
it is easy to perceive that in consequence of the equation (18) the quantity (23) will vanish, and therefore the foregoing value of ZT will always satisfy the equation (17).
Having thus a particular value general one by assuming In
fact, there
thence results
H=KHA n J
,
-&Q
of
^
H, we immediately get the
dh a8
ai a2 a3 )
'
,
the two arbitrary constants which the general integral ought to contain being K, and that which enters implicitly into the in-
H
= V" requires that But the condition should vanish when h is infinite, and consequently the particular definite integral.
value adapted to the present investigation
H = K.H.[ t
a
is
OF ELLIPSOIDS OF VARIABLE DENSITIES. 11.
The value
of
and
<
207
H being
known, we may readily and p. For we have im-
V
find the corresponding values of
mediately (86) ,
-"
and as the function
<
is
rational
and
entire,
and the
partial dif-
V relative to h is finite, it follows that all the partial differentials of V are finite and consequently, by what precedes
ferential of
;
(No. 7) the condition (9') V, as well as the equation equations
and
(b)
(c)
dV
by the foregoing value of = V". Hence the
is
satisfied
(2)
and condition
No. 6 will give, since
-^
and h must be supposed equal
1
*
dV dV
to zero in these equations,
_dV since
where h =
now we
If
n
s
+l
since
is
0,
ar
= a/;
and therefore /22
substitute for
always positive,
clear
it is
V
its
we
value (26), and recollect that
get
from the form of
H
Q
that this quantity
may
always be expanded in a series of the entire powers of h*. In the preceding expression, (27), H^ indicates the value of
H
when h =
and
<' the corresponding value of
or that
which
would be obtained by simply changing the unaccented
letter
?2 from ,
0,
into the accented ones
,
fry)
'=<';
/,
< = <&';
f2',
<
f/ deduced
*.'=.''.
ON THE DETERMINATION OF THE ATTRACTIONS
208
now be
It will
easy to obtain the value of
V
correspond-
ing to 1
'jfo'X, without integrating the formula racteristic of
any
and
rational
1,
where In
entire function.
from what precedes (No. in a finite series of the form
to see
F
No.
(1)
that
9),
is
the cha-
fact it is
2 ',
p'
= v\
{
easy
+ 6.# + &c.
&c. have been replaced with their values
a?
Hence, we immediately
By
F
...(28)
we may always expand 5,fc'
after a?/,
...... a. )
(7).
get
+ b& + b& + &c.} ............. (29).
b.ti
comparing the formulae
(2G)
and
(27) it is clear that
any
term, as & r $/ for instance, of the series entering into p, will have for corresponding term in the required value of V, the
quantity 2
y
,
,
,
'
j5T
"
g
being a particular value of
,
h-"dh
.
H satisfying
and immediately deducible from
<
by
the equation (17) the method before ex-
plained.
12.
All that
now
remains,
is to
V'V0 = VV'<
demonstrate that ........................ (31),
whatever may be. For this purpose let us here resume the value of y<, as immediately deduced from the equation (2") No.
7, viz.
_
trap
.
...(32),
OP ELLIPSOIDS OF VARIABLE DENSITIES.
209
where for simplicity the indices at the foot of the letters ( and a have been omitted, and their accents transferred to the letters themselves. Moreover all the finite integrals are supposed taken from
1 to s
+
1.
in the last expression we immediately get to prevent ambiguity, we write br in a moment, V'^>, the place of the original a r and omit the lower indices as before,
By making and
h
if for
'
we
obtain
where to avoid all risk of confusion r has been changed into r", and the double accent of this index transferred to the letters f and b themselves.
We
will
now
conceive the expression (32) to be written in
the abridged form
the order of the terms remaining unchanged. If then
we
recollect that the accents
have no other
office to
perform than to keep the various finite integrations quite disfinal results they may be tinct, and consequently that in the
permuted in any way at
will,
we
shall readily get
14
ON THE DETERMINATION OF THE ATTRACTIONS
210
all
the finite integrals being taken from
from/ = l to/ = 5 +
rltor = s+l
9
and
l.
In order to obtain the required value
we
clear that
it is
shall only
have
to
add the
first
of the five
preceding quantities to the sum of the four following ones multiplied by h'j and to render this more easy, we have appended to
each of the terms in the preceding quantities a number inclosed in a small parenthesis.
Now
since the accents
have likewise a2 (1), (6)
and
(2), (3), (7)
= V + h?,
be permuted at will, and we easy to see that the terms marked
may
it is
mutually destroy each other. In like manner, (18) mutually destroy each other; the same may
(12)
and
evidently be said of (13) and (16), of (15) and (17), of Moreover, the four quantities (19), and of (8) and (14).
and
(10)
(11) will
Hence the
(9)
(4), (5),
do so likewise, and consequently, we have
truth of the equation (31)
is
manifest.
and
OF ELLIPSOIDS OF VARIABLE DENSITIES.
211
Application of the preceding General Theory to the Determination of the Attractions of Ellipsoids. 13.
erted
required to determine the attractions exa', &', c whether the
it is
Suppose
by an
ellipsoid
whose semi-axes are
attracted point p is situated within the ellipsoid or not, the law of the attraction being inversely as the n' th power of the distance. Then it is well known that the required attractions may always
be deduced from the function
p'dx'dydz
~j< p being the density of the element dxdy'dz of the ellipsoid,
and
x, y, z
being the rectangular co-ordinates of p.
We
may avoid the breach of the law of continuity which takes place in the value of F, when the pointy passes from the interior of the ellipsoid into the exterior space, by adding the positive quantity
w2
to that inclosed in the braces,
wards suppose u evanescent
now
in the final result.
and may
after-
Let us therefore
consider the function
_
pdxdydz
f
this triple integral like the preceding including all the values
of x'
9
y',
z, admitted
by the condition y" z tF*V**'is*
*1 If
now we suppose
V
2
the density p 7/
2
-i-
which will simplify f(x, y and then compare
g'
,
2
is
of the form
'\w 2
/ KV'^
z'}
when p
is
(34)>
constant and
n
this value with the
= 2,
one immediately deducible from the general expression (28) by supposing for a moment
n
= n,
viz.
142
ON THE DETERMINATION OF THE ATTRACTIONS
212
we
see that the function
than F.
But
always be two degrees higher
/ will
since our formulae
proportion as the degree of
F
is
become more complicated higher,
in
will be simpler to
it
determine the differentials of V, because for these differentials the degree of and /is the same. Let us therefore make
F
-n
^
p'(x-xJ}dxdydz
ax
J
<
(x
_^
_ y
then this quantity naturally divides
(z
_ z 'Y + u
itself into
two
parts, such
that
*
pdxdydz'
A'
where
_
\(x-
A" =
and
r
{(x
By that
n
- xj +
(y
- y'Y + (s - z'Y +
comparing these with the general formula 1 = n + 1, and consequently n = ri + 2.
2
2 J
(1), it is clear
In this
way
the expression (28) gives
which coincides with
The simplest and then by No.
(34)
by supposing F=f.
case of the present theory 11,
we have $ '=
1
quantity required, and as the general
where /(#', y', z) = 1,
= 1, when
^4' is
the
No.
11, then from the obtain immediately
series (29),
its first term, we the value of A' following, (30),
reduces itself to
formula
is
arid 5
A'=
(35),
-^a'b'c'l'-!^
because in the present case
H=
1,
5
=
3,
and n
= ri +
2.
OF ELLIPSOIDS OP VARIABLE DENSITIES.
2J3
Again, the same general theory being applied to the value of
A"
given above,
F(x' y\ z) 9
and hence by No.
we
get
= - x'f(x'
= -x' (when/= 1), F(x y z} = a'f> In this way 9
y\
z'}
f
11,
y
,
series (29) again reduces itself to a single terra, in
and
&'=',
H
and the particular value the superfluous constant
,
= -',
corresponding thereto, by omitting
Q
y
*
the
which
,, 2
2
~r
(Q>
-^ -T G
will be (No. 10),
)
These substituted in the general formula
(30) as before,
imme-
diately give
r/
A,> A"
,
/-
dh7
2
and consequently by reduction since
The
f = x,
value of ^4 just given belongs to the density
Hence we immediately obtain without responding values 1
J
\-ri
dV
~ dz
calculation the cor-
ON THE DETERMINATION OF THE ATTRACTIONS
214 If
now we
suppose moreover
1 D ^ !-'
r n'+l
du
J
the method before explained (No.
271-1
(1) ,
if for
will immediately give
1 1)
T
ri
and therefore
J
2
'
.
+ ,
", ,
abcu
abridgment we make
*m the total differential of
V may
'
be written
which being integrated in the usual way by first supposing h and then completing the integral with a function of h, to be afterwards determined by making every thing in Fvariable, constant,
we
get
abc
J
x abc
k being a quantity absolutely constant, which is equal to zero when ri> 1. What has just been advanced will be quite clear if we recollect that h may be regarded as a function of x, y, z and u, determined by the equation
seeing that
2
2
= a' + h\ V = b + Ji\ 2
f
*
and
2
c
= c' + tf. 2
OF ELLIPSOIDS OF VARIABLE DENSITIES. After what precedes,
seems needless to enter into an ex-
it
V belonging
amination of the values of density p, since
it
equally applicable
215
must be
to other values of the
method
clear that the general
is
when
where /is the characteristic of any rational and entire function.
The
quantity
A
when we make u =
before determined
0,
serves to express the attraction in the direction of the co-ordinate x of an ellipsoid on any point p, situated at will either within or
without
But by making u =
it.
1
and
it
is
=
x*
if
+
we have
in (37)
z2
+
thence easy to perceive that
(?
+
*
*
.......... (38) '
when p
is
within the
tion (38)
h must constantly remain equal to zero, and the equawill always be satisfied by the indeterminate positive
quantity
-5
ellipsoid,
2 .
When
on the contrary
no longer remain equal
to zero,
p
exterior to
is
it,
h can
but must be such a function of
x, y, z, as will satisfy the equation (38), of
which the
last
term 2
now
evidently vanishes in consequence of the numerator o and all remain Thus the forms of the quantities A, B, C, .
D
V
unchanged, and the discontinuity in each of them
falls
upon
the quantity h.
To compare
the value of
by the ordinary methods,
we
A
here found with that obtained
shall simply
have
to
make ri=2
in
= i\V. In this way
hdk -^- = -
f
A = -4:7rabcxl ,
7
,
,
J^ a be
da
,,,
,
,-,,
4?ra b c
,
x
da
[ J^ a be I
da
w
-
ON THE DETERMINATION OF THE ATTRACTIONS
216
But the
may
easily be put under the form of
a definite integral,
by writing
- in the place of a under the
sign of integration, will result
and again inverting the
last
quantity
Thus
limits.
there
A= which agrees with the ordinary formula, since the mass of the .,
.
.
^irab'c
ellipsoid is
,
and a
=
,
a
2
7o
+
h
.
Examination of a particular Case of the General Theory exposed in the former Part of this Paper. 14. There is a particular case of the general theory first considered, which merits notice, in consequence of the simplicity of the results to which it leads. The case in question is that where we have generally whatever r may be
=a
/
ar
/
.
Then
by
the equation (19) which serves to determine = k. a 2 supposing k
If
now we employ
= p cos 0J,
f2 = p
becomes
a transformation similar to that used in
obtaining the formula (14), No.
fj
(/>,
sin
Ol cos
and then conceive the equation
6,
by making
2,
f3
(39)
= p sin 0,
sin
8),
cos
3,
&c.
deduced from the condition
that
must be a minimum (vide No.
2
we
shall
have
OF ELLIPSOIDS OF VARIABLE DENSITIES. 1
d^... d&^p"- sin
6^
2
0^... s
sin
2
dp)
now
sin
2 Sfr =
1
a
0* sin
2
...
sin 0V-i
'
2
1
p
.
manner before explained (No. the equivalent of (39) by reduction
Proceeding obtain for
l
p
and
217
in the
2
(b = d-^-T 4
2
s
1
8),
we
dd)
np
-p>) 'dp 3
!'^. k
*ffr d0
a
But
P
this equation
may
be
satisfied
by a
function of the form
being a function of p only, and afterwards generally S r a In fact, if we substitute this value of in r only.
function of (40),
and then divide the
satisfied
by
result
by
clear that
it is
0,
the system
"T ,
o
CQS ""=
0J-, ~7\
&c.
&c.
^-3 7?i
7=\
,
"
X*- 2 /12
&c.
&c.
combined with the following equation,
d *P
+ s-i-np*
p(l-p*) 'Pdp
Pdp* where
k,
dP
\, X2 X3 &c. ,
,
+ A, p*
k_ l-p*~
are constant quantities.
_>/v
it
will be
ON THE DETERMINATION OF THE ATTRACTIONS
218
In order
to resolve the
system
(41), let
us here consider the
general type of the equations therein contained, viz.
_ d*^ + dB^r
Now
if
we
d^
cos 68_ r l)}
(r
p
is
\,.y+1
+
d0 _r
T
(sin G\.r
8
on the nature of the results obtained in
reflect
a preceding part of this paper, -r is of the form
where
/
r
'
sin 6s.r
will not be difficult to see that
it
= cos# _r
a rational and entire function of ^
8
,
and
i
a
whole number.
By
substituting this value in the general type and
Xwl = -
we
(*'+*
making
-2) ..................... (43)
readily obtain
To
assume
satisfy this equation, let us
substituting in the above and equating separately the coefficients of the various powers of ^, we have in the first place
Then by
from the highest
\^ = -e(e + r-l) ..................... (44), and afterwards generally
~
"e
<+'
But the equation (44),
by writing
2t
(43)
i (r} for
+
may ,
It e
i
i
.
2
x 2e
evidently be
and
2t
1
.
+ - 2t - 3
*'
r
made
i(r+1} for e, since
to coincide with
then both will be
comprised in
-"'
)
!*'"'
Hence we (41),
+ -2) .................. (45). '-
readily get for the general solution of the system
219
OF ELLIPSOIDS OF VARIABLE DENSITIES. JM
H-D
-
f j
t
- I'M -1}
<>} {i^
______ )
where
/-t
= cos # _ g
ever, provided
i
r,
(r)
and
^
._._
(,)
4
_
'
5}
i (r) represents
any
what-
positive integer
i (r+l} .
never greater than
is
Though we have
thus the solution of every equation in the first maybe obtained under a simpler
system (41), yet that of the form by writing therein for
We
+7
Jrtu
*
X^
its
i (2}
value
deduced from
shall then immediately perceive that it is satisfied
(45).
by
cos
In consequence of the formula
n_. 1
2
which
the equation (42) becomes ,
~~"
/^
'
2\
p(l-p*)
dp
(45),
is satisfied
*
~7
\
dp
{
2
'
by making k =
\
(8)
(i
i
1-p
p
(8)
+2a))(i
-}-
2co+n
1),
and
-2 where
co
X 2/
represents
(8)
+ 2&) + s - 2
any whole
.
2i (8}
positive
+ 2&) + ^ -
4
2W _4 __
CC
'
number.
Having thus determined all the factors of $, it now only the remains to deduce the corresponding value of H. But in be differential will the value H, equation satisfying particular had from (/> by simply making therein
H
,.
_
since in the present case
Hence,
it
is
we have
generally a r
'
= a.
clear that the proper values of
be here employed are
all
1?
2,
#3 &c. to ,
constant, and consequently the factor
ON THE DETERMINATION OF THE ATTRACTIONS
220
entering into
is
likewise constant.
factor as superfluous,
Pa
Neglecting therefore this
get for the particular value of H,
-
since
and
we
represents
what
P becomes when p is
of
Substituting this value of 2 2 2 there results since a = a' + h
H
changed into a
in the equation (25),
= K.P,
,
No.
.
,
10,
.................. (46),
K being an arbitrary constant quantity. Thus the complete value number
sidered in the present
of
V for
the particular case con-
is
(47),
and the equation value of
(27),
No.
11, will give
for the
corresponding
p',
p=
,
nss+1
rf" \
2
where P/, %^ @ 2 &c. are the values which the functions P, &c. take when we change the unaccented variables t 2 ',
,
fi
,
f2 '*--ft
m
*
tne con-esponding accented ones f/, f2', ...f/,
and
n
or the value of
P when =
1
/o
i is written in the place of
;
(8)
*
.
where as well as
in
what follows
OF ELLIPSOIDS OF VARIABLE DENSITIES.
The
differential equation
which serves
to determine
we
introduce a instead of h as independent variable, present case be written under the form
- 2) a' 2
(i
221
+ 2w) (i + 2a> + n - 1
)
If when
may
in the
a2 } H,
and the particular integral here required is that which vanishes when h is infinite. Moreover it is easy to prove, by expanding in series, that this particular integral is
we make
provided
the variable r to which
have been
after all the operations
But
the constant
k'
may
Aw
refers vanish
effected.
be determined by comparing the
power of a in the expansion of the last formula with the like coefficient in that of the expression (46), and thus we have coefficient of the highest
/I/
ii-VO
*-
I
_
/
_
_
2. 4. 6.
Hence we rr
T>nn = P0 e i
...
readily get for the equivalent of (47),
2 ...e,_ 1
x
..2ft)
x
n4-2^+2ft)-l.w + 2/+2ft)+l...w+2i+4ft)-3 2
.
4.
6..
Ka i+2w (- l) wa Aw a2r f^daa ^'^ i
1
.2ft,
2
1
(a
- a' ) 2
In certain cases the value of Fjust obtained will be found more convenient than the foregoing one (47). Suppose for instance we represent the value of V when h = 0, or a = a' by F .
Then we
shall
hence get
2
.
4. 6.. .2ft)
ON THE ATTRACTIONS OF ELLIPSOIDS.
222
which
in consequence of the
well-known formula
~~ by reduction becomes
r
since in the formula
end of the
By become
(8),
r ought to be made equal to zero at the
process.
conceiving the auxiliary variable u to vanish, it will clear from what has been advanced in the preceding
number, that the values of the function V within circular planes and spheres are only particular cases of the more general one We have 2 and 5 = 3 respectively. (49), which answer to s thus by combining the expressions (48) and (49), the means of
V
when the density p is given, and vice versa; and the present method of resolving these problems seems more simple if possible than that contained in the articles (4) and (5) determining
of
my
former paper.
ON THE MOTION OF WAVES IN
A VARIABLE CANAL OF SMALL DEPTH
AND WIDTH*
From
the Transactions of the Cambridge Philosophical Society, 1838.
[Read
May
15, 1837.]
ON THE MOTION OF WAVES IN A VARIABLE CANAL OF SMALL DEPTH AND WIDTH. THE equations and conditions necessary for determining the motions of fluids in every case in which it is possible to subject them to Analysis, have been long known, and will be found in the First Edition of the Mec. Anal, of Lagrange. Yet
the difficulty of integrating them
is
such, that
many
of the most
important questions relative to this subject seem quite beyond the present powers of Analysis. There is, however, one particular case which admits of a very simple solution. The case in question is that of an indefinitely extended canal of small breadth and depth, both of which may vary very slowly, but in other respects quite arbitrarily. This has been treated of in the following paper, and as the results obtained possess considerable simplicity, perhaps they may not be altogether unworthy the Society's notice.
The
general equations of motion of a non-elastic fluid acted (g) in the direction of the axis z, are,
on by gravity
supposing the disturbance so small that the squares and higher powers of the velocities &c. may be neglected. In the above formulae
p=
=
pressure, p density, and < is such a function of that the velocities of the fluid particles parallel to t, the three axes are
x, y, z
and
M== (d$\ T^ \dxj
dp >
Tz.
15
ON WAVES IN A VARIABLE CANAL
To ditions
the equations (1) and (2) relative to the exterior
requisite to add the consurfaces of the fluid, and if of these surfaces, the corre-
it is
A=
be the equation of one sponding condition is [Lagrange, Mec. Anal Tom. u.
p.
303.
(i.)]_
dA dA dA = dA W. 77 +-J-W+ -= V+-jdt dx dz dy
Hence .
v
dA d6 = dA 77 + -7
dA d6
,
'
-^ dx dx
dt
The equations
(1)
f
T~ Vdy dy
and
.
(2)
+
dA d6 7-
dz
.
,
,
A
~r (wnen^L ^ dz
_N
/AX = UJ...(A).
with the condition (A) applied to suffice to determine
each of the exterior surfaces of the fluid will
in every case the small oscillations of a non-elastic fluid, or at least in those where
udx is
an exact
4-
vdy + wdz
differential.
In what follows, however,
we
shall confine ourselves to the
consideration of the motion of a non-elastic fluid,
when two
of
the dimensions, viz. those parallel to y and 2, are so small that $ may be expanded in a rapidly convergent series in powers of y and 0, so that
Then plane of
if
we
(a?,
take the surface of the fluid in equilibrium as the #), and suppose the sides of the rectangular canal
symmetrical with respect to the plane (x, z), will evidently contain none but even powers of y, and we shall have
Now
if
represent the equation of the only satisfy one of them as
two
sides of the canal,
we need
OF SMALL DEPTH AND WIDTH. since the other will then be satisfied
powers of y from
The
by
227
the exclusion of the odd
.
A =y
equation (A) gives, since here
Similarly, if z
=
yx
is
/3,
the equation of the bottom of the
canal,
A Q If
= dd>
moreover z
dy -,.,
.......
=
.
.
d(f>
(
when
= 7) ............. (J).
*
be the equation of the upper
surface,
,<*_##_# dz dx dx dt ,
But here^ =
;
/.
also
Substituting from
by
(2)
(3) in (b)
^
#? = dt
we
get
or neglecting quantities of the order
-* Similarly
and
(c)
(a)
becomes
becomes, since f
of the order of the disturbance,
is
-*-s 2
or neglecting (disturbance) z
provided as above
we
" j
2
neglect (disturbance)
Again, the condition coefficients of
=
(2)
gives
by equating
powers and products of y and
=
^ **^ dx'
=
&c.
+
.
+6
separately the
z,
"I
I
,
J
152
ON WAVES
22B If (2'),
now by means
of
(a'),
(c)
(&'),
we
eliminate
$"
from
there results
_, ~
It
A VARIABLE CANAL
IN
now
dx2
+
.
+ ,
ydx) dx
\J3das
"
gy\ df
)
only remains to integrate this equation.
we
shall suppose ft and 7 functions of x which vary very slowly, so that if written in their proper form we
For
this
should have
= ^(a>x), where w
is
a very small quantity. 'd&
^=
Hence a)
2 ,
if
we
and assume,
where A
is
v
,,,
d*ft
co^fr (o)^),
-j-g
Then, = eo,., ^ (cox),- &c.
allow ourselves to omit quantities of the order to satisfy (4),
a function of
x
of the
same kind " ;: ''
omitting
.
2
(g),
V
"
as ft
and
we
7,
have,
;
1,
l
dx
dx j
^
Substituting these in order
2
eo
,
we
(4),
and
neglecting, quantities of the
still
get
91
*AdX dx
-fa
+ (dp d
t
OF SMALL DEPTH AND WIDTH,
now
equating
229
separately the coefficients of /' and /",
dA
dx*
d$
*
~
we
get
dy
dx
The
first,
integrated, gives
dx '
and the second k
= -r- A 8v = 2
,
.
<\/gy
Hence
if
we
by
is,
(v
dx
k V^, the gene-
A=
(c'),
and the actual velocity of the the axis of x,
V^r
neglect the superfluous constant
ral integral of (4)
therefore,
.
fluid particles in the direction of
is
dx
neglecting quantities which are of the order those retained. If the initial values of f and
u
are given,
(ay)
compared with
we may
then deter-
mine /' and F', and we thus see that a single wave, like a pulse of sound, divides into two, propagated in opposite directions. Considering, therefore, only that which proceeds in the direction of
x
positive,
we have
ON WAVES IN A VARIABLE CANAL, &C.
230
Suppose now the value of F' (x) = 0, except from x = a to and x to be the corresponding length of the wave, we have
x
= a + a,
= a + a, t- [dx -j=,
Wgy
,
and
[
,
t
I
dx r=
Sx
= =a
JVffy
,
very nearly.
V<77
Hence the variable length of the wave
is
............ (7).
Lastly, for
any particular phase of the wave, we have .
t
dx -==
= const.
:
therefore
is the velocity with which the wave, or more strictly speaking the particular phase in question, progresses.
From
(5),
(6),
(7),
and
variable breadth of the canal
f = height
u
-j- =5
we
and 7
wave
see that if its
ft'
depth,
@~*
actual velocity of the fluid particles
dx = length and
of the
(8)
of the
wave
7^
velocity of the wave's motion
=
represent the
ON THE REFLEXION AND REFRACTION OF SOUND.
From
the Transactions of the Cambridge Philosophical Society, 1838.
[Read Dec.
11, 1837].
ON THE REFLEXION AND REFRACTION OF SOUND. THE
object of the communication
which
I
have now the
honour of laying before the Society, is to present, in as simple a form as possible, the laws of the reflexion and refraction of sound, and of similar phenomena which take place at the surface of separation of any two fluid media when a disturbance is propagated from one medium to the other. The subject has already been considered by Poisson (Mem. de VAcad., &c. Tome x. p.
The method 317, &c.). one that he has used on
employed by
this celebrated analyst is
many occasions with great success, and which he has explained very fully in several of his works, and recently in a digression on the Integrals of Partial Differential In this way, Equations (TMorie de la Chaleur, p. 129, &c.). is made to depend on sextuple definite integrals. Afterwards, by supposing the initial disturbance to be confined to a small sphere in one of the fluids, and to be everywhere the same at the same distance from its centre, the formulae are
the question
made
depend on double definite integrals from which are ultimately deduced the laws of the propagation of the motion at great distances from the centre of the sphere originally disto
;
turbed.
The chance
of error in every very long analytical process, it becomes necessary to use Definite
more particularly when
Integrals affected with several signs of integration, induced me to think, that by employing a more simple method we should
possibly be led to some useful result, which might easily be overlooked in a more complicated investigation. With this
impression
I
endeavoured to ascertain
how
a plane
wave
of
and refracted waves, accompanied by would be propagated in any two indefinitely extended media of
infinite extent,
its reflected
234
ON THE REFLEXION AND REFRACTION OF SOUND.
which the surface of separation
in a state of equilibrium should
also be in a plane of infinite extent.
suppositions just made simplify the question extremely. may also be considered as rigorously satisfied when light
The They
is reflected.
to the
In which case the unit of space properly belonging
problem
is
\ = ^-^r^.
a quantity of the same order as
inch,
and the unit of time that which would be employed by self in
sound
passing over this small space.
Very
light itoften too, when
these suppositions will lead to sensibly correct this last account, the problem has here been con-
is reflected,
On
results.
sidered generally for all fluids whether elastic or non-elastic in the usual acceptation of these terms ; more especially, as thus its solution is not rendered
One
more complicated.
result of our
so simple that I may perhaps be allowed to mention analysis it here. It is this : If be the ratio of the density of the reflectis
A
B
medium to the density of the other, and the ratio of the cotangent of the angle of refraction to the cotangent of the angle of incidence, then for all fluids ing
the intensity of the reflected vibration the intensity of the incident vibration If
of
now we apply this water, we have
still
B < J.
__ "~
A B A +B
to the reflexion of
sound
A > 800,
maximum
and the
'
at the surface
value of
intensity of the reflected wave will in every case be sensibly equal to that of the incident one. This is what we should naturally have anticipated. It is however noticed
Hence the
here because
M. Poisson has inadvertently been
led to a result
entirely different.
When medium,
is
the velocity of transmission of a wave in the second greater than that in the first, we may, by sufficiently
increasing the angle of incidence in the first medium, cause the refracted wave in the second to In this case the disappear. change in the intensity of the reflected wave is here shown to be that, at the moment the refracted wave disappears, the intensity of the reflected becomes exactly equal to that of the incident one. If we moreover suppose the vibrations of the inci-
such
dent wave to follow a law similar to that of the cycloidal pendu-
ON THE REFLEXION AND REFRACTION OF SOUND. lum, as
is
usual in the Theory of Light,
it
is
235
proved that on
farther increasing the angle of incidence, the intensity of the reflected wave remains unaltered whilst the phase of the vibra-
tion gradually changes. The laws of the change of intensity, and of the subsequent alteration of phase, are given here for all media,
When, however, both the media are elastic, remarkable that these laws are precisely the same as those
elastic or non-elastic. it is
for light polarized in a plane perpendicular to the plane of inci-
dence.
when,
Moreover, the disturbance excited in the second medium, it ceases to transmit a wave
in the case of total reflexion,
in the regular
way,
is
represented
by a quantity
This
factor is a negative exponential.
of which one
factor, for light, decreases
with very great rapidity, and thus the disturbance gated to a sensible depth in the second medium.
is
not propa-
Let the plane surface of separation of the two media be taken as that of (yz\ and let the axis of z be parallel to the line of intersection of the plane front of the wave with (yz}> the axis of x being supposed vertical for instance, and directed downwards ; and A f are the densities of the two media under the then, if
A
constant pressure
P and
s,
^
the condensations,
we must have
= density in the upper medium, = density in the lower medium. (P (1 -f As) = pressure in the upper medium, \P (1 + A^ = pressure in the lower medium. [A (1 -f *) (A l (l +*,)
Also, as usual, let
<j>
be such a function of x, y,
resolved parts of the velocity of axes,
may
any
z,
that the
fluid particle parallel to the
be represented by
dx
'
'
dy
dz'
In the particular case, here considered, $ will be independent of z, and the general equations of motion in the upper fluid will be
ON THE REFLEXION AND REFRACTION OF SOUND.
236
where we have
P4
,
~ '
'A or
s
by eliminating
Similarly, in the lower
medium
where
PA The above tion,
fluids
which must be ;
particles
known
general equations of fluid
mo-
satisfied for all the internal points of
both
are the
but at the surface of separation, the velocities of the perpendicular
must be the same
for
and the pressure there Hence we have the particular
this surface
to
both
fluids.
conditions
d$^d$,
dx~ dx As = A s t
.
]
j, t
= 0),
(where x
}
neglecting such quantities as are very small compared with those retained, or
by eliminating dx
5
and
s t
,
we
get
dx (A).
The general equations (1) and (2), joined to the particular conditions (A) which belong to the surface of separation (yz) 9 only, are sufficient for completely determining the motion 6f our two fluids, when the velocities and condensations are independent
of the co-ordinate
z,
whatever the
We shall not here attempt to
initial
disturbance
may
give their complete solution,
be.
which
would be complicated, but merely consider the propagation of a plane wave of indefinite extent, which Js accompanied by its reflected and refracted wave.
ON THE REFLEXION AND REFRACTION OF SOUND.
237
all the particles, in any front of the the same at the same instant, we shall
Since the disturbance of incident plane wave, have for the incident
is
wave
retaining b and c unaltered, reflected and refracted waves,
we may
give to the fronts of the
any position by making
for
them
=F Hence,
we have
in the upper
medium,
+ F(a'x + by + ct)
=f(ax+by+ct)
........... (4),
and in the lower one <=f,(a
t
v+fy + ct)
................................ (5).
These, substituted in the general equations
(1)
and
(2),
give
(6).
=
Hence, a
a,
wh^re the lower signs must evidently be taken
This value proves, that the to represent the reflected wave. is equal to that of reflexion. of incidence In like manner, angle will give the known relation of sines for the incirefracted wave, as will be seen afterwards.
the value of
dent and
a,,
Having satisfied the general equations (1) and (2), it only remains to satisfy the conditions (A), due to the surface of sepaBut these by substitution give ration of the two media.
af because a
=
Hence by
(by
a,
+ ct) - aF (by + and x =
writing,
ct)
=
af (by + t
ct)
,
0.
to abridge,
the characteristics only of
the functions 2
VA
'
y
w,
ON THE REFLEXION AND REFRACTION OF SOUND.
238 or if
we
introduce
the angle of incidence and refraction,
6, 0,,
since
= &F
/i
COt
"
y
2
VA
cot 6
= l/A,_cot0
^,
cotfl
A,
and therefore
/'~A, + cot0,' A cot 6 which exhibits under a very simple form, the intensities
of the
in
disturbances,
ratio
between the
the incident and reflected
wave.
But the equations
(6)
give
and hence
7
7,
sin 6 t
sin
'
the ordinary law of sines.
The
reflected
wave
will vanish
A
=
when
cot
I?
A~^tl
;
which with the above gives
=AA/
cot
V
V (7
.r
7
A / A M-( A 7) A)
Hence the reflected wave may be made to vanish if 7* y* and (vA) 2 (%A,) have different signs. For the ordinary elastic fluids, at least if we neglect the change of temperature due to the condensation, of the nature of the gas, and therefore
A
A
or
A
is
independent
ON THE REFLEXION AND REFRACTION OF SOUND.
239
Hence tan 6
=2 7,
which
is
the precise angle at which light polarized perpendicular wholly transmitted.
to the plane of reflexion is
But it is not only at this particular angle that the reflexion of sound agrees in intensity with light polarized perpendicular to For the same holds true for every angle the plane of reflexion. of incidence.
In
since
fact,
2
7
A,
2A -
and the formulae
(7)
give 2
sin J?_
f
2
sin 6
=
6
~
2
tanfl
6,
tan
sin 0,
tan
" sin
sin
2
tan
2
0,
_ tan (0-0)
0" -tan (0+0,)' /
the same ratio as that given for light polarized perpendicular to the plane of incidence. (Vide Airy's Tracts, p. 356)*.
which
is
What is plane.
precedes is applicable to all waves of which the front In what follows we shall consider more particularly
the case in which the vibrations follow the law of the cycloidal
pendulum, and therefore in the upper medium we shall have,
= a sin (ax + by + ct) + ft sin
(j>
(
ax
+ by + ct)
......... (8).
Also, in the lower one, <,
and as
= a,
sin (a tx
+ ly + ct)
this is only a particular case of the
:
more general
one, be-
fore considered, the equation (7) will give
y >
7, or the velocity of transmission of a wave, be in lower than in the upper medium, we may by dethe greater a render a y imaginary. This last result merely indicates creasing
If
/
that the form of our integral *
must be changed, and that as
[Airy on The Undulatory Theory of Optics,
p.
Ill, Art. 129.]
far as
ON THE REFLEXION AND REFRACTION OP SOUND.
240
regards the co-ordinate x an exponential must take the place of In fact the equation, the circular function.
may
be
satisfied
by
= e-
< y
(where, to abridge, ty
when
this is
(A) due
done
it
is
for
put
by
+ ct)
provided
will not be possible to satisfy the conditions without adding constants to
to the surface of separation,
the quantities under the circular functions in therefore take, instead of (8), the formula,
$ = a sin
(ax 4 by
Hence when x -^U&
+ ct + e) + ft sin
= 0, we
= aa cos
- = ca cos
ax + by +
(
.
ct
get
(ijr
(^r
+ e)
+ e)
(^ +
aft cos
cos (^r
c
e^),
+ e,),
these substituted in the conditions (A), give
e)ft cos
(-^
+ e,) =
a cos fy + e)+ft cos
(<\fr
+e =
a cos ty +
t
^ ^ sin ^, -
)
B cos ^
these expanded, give
a cos a sin
e
e,
/3
cos e t
= 0,
+ ft sin e = --
a.
cose-}- ft cos e
a sin e
t
-r-'
+ ft sin e = 0. /
-
a
t
J5,
B,
;
We +e
t
)
must
....
(9).
ON THE REFLEXION AND REFRACTION OF SOUND. Hence,
we
241
get
2a sin
e
=
2a cos e =
2/3 sin e
t
a -
B,
B,
and, consequently,
and tan
This
e
= we would apply it elastic, we have, because
result is general for all fluids, but if
which are usually
to those only
in this case
2
7
A = y* A,
called
,
But generally (11);
and
therefore,
because
As
^
by
,
7
substitution,
and a
= tan 0.
=
e we see from equation (9), that 2e is the change which takes place in the reflected wave and this is precisely the same value as that which belongs to light polarized
e
t ,
of phase
;
perpendicularly to the plane of incidence ; (Vide Airy's Tracts, thus see, that not only the intensity of the reflected p. 362*.) wave, but the change of phase also, when reflexion takes place at
We
the surface of separation of two elastic media, same as for light thus polarized.
is
precisely the
Airy, ubi sup. p. 114, Art. 133,
16
ON THE REFLEXION AND REFRACTION OF SOUND.
242
As
a
= /3, we
see that
when
there
is
no transmitted wave the
intensity of the reflected wave is precisely equal to that of the This is what might be expected : it is, however, incident one.
noticed here because a most illustrious analyst has obtained a different result. (Poisson, Memoires de V Academic des Sciences,
Tome
X.) arrives at is,
The
result
which
That
at the
moment
celebrated mathematician
this
the transmitted
wave
ceases to
the intensity of the reflected becomes precisely equal to On increasing the angle of incidence that of the incident wave. this intensity again diminishes, until it vanish at a certain exist,
On still farther increasing this angle the intensity conangle. tinues to increase, and again becomes equal to that of the incident wave,
when
the angle of incidence becomes a right angle.
It may not be altogether uninteresting to examine the nature of the disturbance excited in that medium which has ceased to
For
transmit a wave in the regular way. resume the expression,
= Be~* ' sin ty = Be~a :
,
or
if
we
substitute
equations (10)
;
and
for
B,
its
for a/, its
''
x
this purpose,
sin (by
+ ct)
we
will
;
value given by the last of the value from (11) this expression, ;
in the case of ordinary elastic fluids reduce to
where
7*
A=7
2 ,
A,
,
will
-1
cos e
.
e
sn
A-
/
+ c,
X being the length of the incident wave measured perpendicular to its own front, and 6 the angle of incidence. We thus see with what rapidity
in the case of light, the disturbance diminishes as the surface of separation of the two media
the depth
x below
increases
and
;
becomes less as and entirely ceases when 6 is
also that the rate of diminution
approaches the critical angle,
exactly equal to this angle, and the transmission of a the ordinary way becomes possible.
wave
in
ON THE LAWS OF
BEFLEXION AND EEFEACTION OF LIGHT AT THE COMMON SURFACE OF TWO NONCKYSTALLIZED MEDIA.
From
the Transactions of the Cambridge Philosophical Society) 1838.
[Bead December n, 1837.]
162
ON THE LAWS OF THE REFLEXION AND REFRACTION OF LIGHT AT THE COMMON SURFACE OF TWO NON-CRYSTALLIZED MEDIA.
M. CAUCHY seems
to
have been the
first
who saw
fully
the utility of applying to the Theory of Light those formulae which represent the motions of a system of molecules acting on
each other by mutually attractive and repulsive forces
supposing always that in the mutual action of any two particles, the particles may be regarded as points animated by forces directed
along the right line which joins them. applied to those
by mechanical
compound division,
This
particles, at least,
seems rather
nomena, those of crystallization
;
last supposition, if
which are separable
restrictive
for instance,
;
as
seem
many
phe-
to indicate
certain polarities in these particles. If, however, this were not the case, we are so perfectly ignorant of the mode of action of the elements of the luminiferous ether on each other, that it
would seem a
safer
method
to take
some general physical princiassume certain maybe widely different from
ple as the basis of our reasoning, rather than
modes of action, which, after all, the mechanism employed by nature
more especially if this ; used as a itself, particular case, those before more simple by M. Cauchy and others, and also lead to a much The principle selected as the basis of the process of calculation. principle include in
reasoning contained in the following paper way the elements of any material system
is this
may
:
In whatever
act
upon each
the internal forces exerted be multiplied by the elements of their respective directions, the total sum for any other,
if
all
assigned portion of the mass will always be the exact differential of some function. But, this function being known, we can immediately apply the general method given in the Mtfcanique Analytique,
and which appears
to
be more especially applicable to
ON THE REFLEXION AND REFRACTION OF LIGHT.
246
problems that relate to the motions of systems composed of an particles mutually acting upon each other.
immense number of
One that
of the advantages of this method, of great importance, is, are necessarily led by the mere process of the calcu-
we
lation, and with little care on our part, to all the equations and conditions which are requisite and sufficient for the complete solution of any problem to which it may be applied.
The present communication is confined almost entirely to the consideration of non-crystallized media; for which it is proved, that the function due to the molecular actions, in its most general
A
and B-, the form, contains only two arbitrary coefficients, values of which depend of course on the unknown internal constitution of the
medium under
for the
and
consideration,
it
would be
most general
case, that any arbitrary disturbance, excited in a very small portion of the medium, would
easy to shew,
two by normal, the other
in general give rise to
spherical waves, one propagated
entirely
entirely
and such that
by transverse,
vibrations,
the velocity of transmission of the former wave be represented by *JA, that of the latter would be represented by *JB. But in the transmission of light through a prism, if
though the wave which incapable
itself
of giving rise
is
propagated by normal vibrations were yet it would be capable
of affecting the eye, to
an ordinary wave
of light
propagated by
transverse vibrations, except in the extreme cases where
~
r
IB
a very l ar e quantity
be regarded as infinite;
may
that the equilibrium of our A.
;
4.
5>3 A and '
,
-g
^
e are therefore
-5=0,
which, for the sake of simplicity,
and
it
is
not
difficult to
medium would be
prove
unstable unless
compelled to adopt the
latter
value of
thus to admit that in the luminiferous ether, the velocity
of transmission
waves propagated by normal vibrations very great compared with that of ordinary light.
The
of
is
principal results obtained in this paper relate to the tensity of the wave reflected at the common surface of two
ON THE REFLEXION AND REFRACTION OF LIGHT.
247
media, both for light polarized in and perpendicular to the plane of incidence ; and likewise to the change of phase which takes
In the former case, our place when the reflexion becomes total. values agree precisely with those given by Fresnel ; supposing, as he has done, that the direction of the actual motion of the particles of the luminiferous ether is perpendicular to the plane But it results from our formulae, when the light
of polarization.
polarized perpendicular to the plane of incidence, that the expressions given by Fresnel are only very near approximations ; is
and that the intensity of the
wave
reflected
will never
become
absolutely null, but only attain a minimum value ; which, in the case of reflexion from water at the proper angle, is yiy part This minimum value increases of that of the incident wave. rapidly,
the index
as
of refraction
increases,
and thus the
quantity of light reflected at the polarizing angle, becomes considerable for highly refracting substances, a fact which has been
long
known
to
experimental philosophers.
proper to observe, that M. Cauchy (Bulletin ctes Sciences, 1830) has given a method of determining the intensity of the waves reflected at the common surface of two media. It
He
may be
has since stated, (Nouveaux Exercises des Mathematiques^]
that the hypothesis employed on that occasion is inadmissible, and has promised in a future memoir, to give a new mechani-
cal principle applicable to this and other questions ; but I have not been able to learn whether such a memoir has yet apThe first method consisted in satisfying a part, and peared.
only a part, of the conditions belonging to the surface of junction, and the consideration of the waves propagated by normal
was wholly overlooked, though it is easy to perceive, waves of this kind must necessarily be produced when the incident wave is polarized perpendicular to the plane of incidence, in consequence of the incident and refracted waves
vibrations
that in general
being in different planes. consideration
of these
last
Indeed,
without introducing the
is impossible to satisfy the whole of the conditions due to the surface of junction of But when this consideration is introduced, the the two media.
whole of the conditions
waves,
may
be
it
satisfied,
and the principles
ON THE REFLEXION AND REFRACTION OF LIGHT.
248
given in the
Mfaamque Analytique became abundantly
suffi-
cient for the solution of the problem.
In conclusion,
it
may
be observed,
that the radius of the
action of the molecular forces has been resphere of sensible with respect to the length X of a wave insensible as garded and thus, for the sake of simplicity, certain terms have of light,
been disregarded on which the different refrangibility of difThese ferently coloured rays might be supposed to depend.
which are necessary
terms,
to
be considered
when we
are
treating of the dispersion, serve only to render our formulae uselessly complex in other investigations respecting the pheno-
mena
of light.
Let us conceive a mass composed of an immense number of molecules acting on each other by any kind of molecular forces, but which are sensible only at insensible distances, and let moreover the whole system be quite free from all extraneous action of every kind. Then a?, y and z being the co-ordinates of any particle of the medium under consideration when in
equilibrium, and
same particle in a state of motion (where are very small functions of the original co-ordi-
the co-ordinates of the w, v,
w
and
we get, ), of any particle and of the time ($)), (a;, y, by combining D'Alembert's principle with that of virtual ve-
nates
locities,
d*u
5.
Su
Dm
and
Dv
+
d*v
~
+
d*w
5,
}
~
being exceedingly small corresponding elements
mass and volume of the medium, but which nevertheless contain a very great number of molecules, and S
of the
ternal actions of the particles of the
Indeed,
if
would be
&
were not an exact
possible,
and we
medium on each
other.
differential, a perpetual *motion have every reason to think, that
ON THE REFLEXION AND REFEACTION OF LIGHT.
249
the forces in nature are so disposed as to render this a natural impossibility.
Let us now take any element of the medium, rectangular in a state of repose, and of which the sides are dx, dy, dz the length of the sides composed of the same particles will in a ;
motion become
state of
dx=dx (1 where
s
l
,
s2 , s3 are
If,
moreover,
a,
/3,
and y
dy=dy (1 +* )>
+*,),
a
dz
=dz
(1
+ss)
exceedingly small quantities of the
first
;
order.
we make,
will be very small quantities of the same order. may be the nature of the internal actions, if we
But, whatever
by
represent
&
dx dy dz,
the part of the second member of the equation (1), due to the molecules in the element under consideration, it is evident, that will remain the same when all the sides and all the
angles of the parallelepiped, whose sides are dx dy dz, remain and therefore its most general value must be of the
unaltered,
form
= function
fo, *, *a , a,
& y]. ~^~
But
5X
,
first order,
y being very small quantities of the we may expand in a very convergent series of
52
,
s3 ,
a,
/3,
<j>
the form
<
,
<
(f) l}
2
,
&c. being homogeneous functions of the six quanof the degrees 0, 1, 2, &c. each of which
tities a, /3, y, $ t , sa , ss
very great compared with the next following one. If now, p represent the primitive density of the element dx dy dz, we may in the formula (1), which write p dxdy dz in the place of
is
Dm
will thus become, since
is
constant,
ON THE REFLEXION AND REFRACTION OF LIGHT.
250
the triple integrals extending over medium under consideration.
But by the system
is
supposition,
in equilibrium,
=
the whole volume
when u = 0,
v
=
and
of the
w = 0,
the
and hence [[(dx dy dz S<^
:
seeing that <>, is a homogeneous function of s 1? s 2 s a a, /5, 7 &c. If therefore we neglect 3 of the first degree only. 4 our which are exceedingly small compared with 2 equation ,
,
<
,
,
,
becomes
\\\pdxdydz \-^% u
+
^
the integrals extending over the whole volume under consideraThe formula just found is true for any number of media tion.
comprised in this volume, provided the whole system be perfectly free from all extraneous forces, and subject only to its own molecular actions.
If
now we can
obtain the value of
<>
2
,
we
shall only
have
to
apply the general methods given in the Mecanique, Analytique. But 2 being a homogeneous function of six quantities of the <
most general form contain 21 arbitrary proper value to be assigned to each will of course depend on the internal constitution of the medium. If, however, the medium be a non-crystallized one, the form of a will remain the same, whatever be the directions of the co-ordi-
second degree, will in coefficients.
its
The
nate axes in space. Applying this last consideration, we shall find that the most general form of for non-crystallized bodies a contains only two arbitrary coefficients. In fact, by neglecting quantities of the higher orders,
it is
easy to perceive that
ON THE EEFLEXION AND REFRACTION OF LIGHT. du
a
and
if
for z
2
dw n P = dx ^j
dz'
medium
is
du ^
T~ dz
> '
7
du dv = TT"+7T'
dx
dy
symmetrical with regard to the plane (xy) w are written will remain unchanged when z and
the
only,
dy
dw
dv
dv = dw -j- + -j-,
251
this alteration evidently changes a and ft to Similar observations apply to the planes (xz) If therefore the medium is merely symmetrical with
But
and w.
a and (yz).
ft.
respect to each of the three co-ordinate planes,
we
see that
2
must remain unaltered when or or
or
In
-z, -w, -a, -ft}
- y, - x, this
u,
way
sulting function
- ft, - 7
\dxj
)
<
w,
a,
ft
y, v,
a,
7
u,
ft}
7.
la;,
the 21 coefficients are reduced to is
and the
9,
re-
of the form
H
O
7
a,
v,
(z,
are written for
>
+1 \dyj
dv dw
~
du dw
n + 2P -j-.-j- +2Q-J-.-Jdz dx dz dy
du dv +2R-J-.-Jdx dy
.
,
=^
2
...
A
.
(A}.
Probably the function just obtained may belong to those crystals which have three axes of elasticity at right angles to each other.
Suppose now we further by making it symmetrical instance.
By
restrict the generality of all
round one
shifting the axis of
x through
angle BO,
becomes
\y
our function
axis, as that of z for
the infinitely small
252
ON THE REFLEXION AND REFRACTION OF LIGHT.
and
,u + v$0
u j
v
w Making will not
[
becomes
J
these substitutions in
remain the same
for the
J
v
I
w
u&0.
we
(^1),
new
see that the form of
axes, unless
and thus we get
under which form
it
may
possibly be applied
to
uniaxal
crystals.
Lastly, if we suppose the function to all three axes, there results
<
2
symmetrical with respect
and consequently,
aw
ON THE REFLEXlUff AND REFRACTION OF LIGHT. by merely changing the two constants and
or,
values of
a, /?,
and
-p((du
dv\*
restoring the
y,
2*,
^+
du fu = --4A (-7-
+
dx \dx
fdu
+
dv
dw\*
J dy
dz)
dw\* ,(d^ +
+
\dz (dv
dw
du
dw\*
dy)
d.w
du dv\] dx dy dy))
. '
'
\dy' dy' dz
This
253
dx' dz
"
~.
*
the most general form that 2 can take for non-crystallized bodies, in which it is perfectly indifferent in what direcis
<
tions the rectangular axes are placed.
The same
result
might
be obtained from the most general value of 8 by the method before used to make $2 symmetrical all round the axis of z, ap<
plied also to the other
axes.
It was, indeed, thus I first
The method given in the text, however, and which similar to one used by M. Cauchy, is not only more simvery
obtained is
two
,
it.
but has the advantage of furnishing two intermediate which may possibly be of use on some future occasion.
ple,
Let us
now
results,
consider the particular case of two indefinitely
extended media, the surface of junction when in equilibrium being a plane of infinite extent, horizontal (suppose), and which
we
shall take as that of (yz) 9
and conceive the axis of x
positive
downwards. Then if p be the constant density of the and p that of the lower medium, 2 and 2 the correupper, due to the molecular actions; the equation functions sponding directed
(1)
<
/
(2)
(/>
adapted to the present case will become
(3);
w
belonging to the lower fluid, and the extended over the whole volume of the being
w
y,
v /}
t
they respectively belong,
triple integrals fluids to
which
ON THE REFLEXION AND REFRACTION OF LIGHT.
254 It
now only remains
to substitute for
to effect the integrations
by
<
and
parts,
2
and
(1) <
2
zero the coefficients of the independent variations. therefore for
value ((7),
its
2
1
fff = - AA HI dx dy dz ^
^
^
JJJ
1
1
Substituting
get
dx dy dz 8 2
(f^U ]
we
their values,
to equate separately to
(-j-
(\dx
+
dv -j-
+ dw\ -j-
fdSu -j
dzj\dx
dy
+
dSv -j-
dy
+ d$W\\ j- r dz
J}
nfffj j j ((&u dv\fdSu d$v\ fdu dw\fdSu d$w\ (dv dw\fdSv dbu + T+-J- -y-+^r" B\ \dxdy dz\ -T-+J-}( -j-+-j- -f -J-+-T- ~r- + ~r~ dx J \dz JJJ IW^ dx)\dy dx) \dz dxj\dz dyj\dz dy
^w
'
o
[
(^
^w
d$v\
+
\\fiy'~d* *"dz'~dy)
=-
d$w
fdu (dx'~dz
^
dw du\ dz'~d^}
d fdu
d
dv
f A fdu + U=j^^-. az .
\dx
(
seeing that
^ = ^00
,
dv-
dw\
dw\ -
-f
J
dz)
dy
we may
2/=co
,
d*v
d*v
d*w
J\d*w + B\ -7-5 +-^-5 1
[_dx*
+
dy
+
(dx'lty
-j-
u
,
fdu d$v
du dv -r^fdv + -r + dw\ r }---2B (-jdx dy dz) \dy
ff Jj
T
+
d zu
+
dv d$u dj/'~d^
dw
d
fdv
_-_(_ d fdu
dw
d fdu
dv\]}
+ ^-.^dz \dx
-7-
*
r^5
dy)]}
neglect the double integrals at the limits = +co; as the conditions imposed at these
2
limits cannot affect the
motion of the system at any finite distance from the origin; and thus the double integrals belong only to the surface of junction, of which the equation, in a state of equilibrium, is
= x.
ON THE REFLEXION AND REFKACTION OF LIGHT. In like manner
+ the since
is
it
we
255
get
triple integral
;
the least value of
x which belongs
to the surface of
junction in the lower medium, and therefore the double integrals belonging to the limiting surface must have their signs changed. If, now, we substitute the preceding expression in (3), equate separately to zero the coefficients of the independent variation $u, &v, Sw, under the triple sign of integration, there results for
the upper
d*u
A
medium
d (du
z z dw\ n (d u d u y')+-5lj-i +-7-5 dz*
dv
P^r=A dx -j--\-j- +-J- + \dx dy ,
,
r dtf
d P r
2
dt
v 2
^ df
.
d fdu
=A-J-. Tdy \dx
dz]
dw\
dv
\dy*
(d*v
d
2
v
+ ^- + ^- +-5 ij-i + TTdz dz)
dz\dx
2
dy
(dx*
dz)
dy
and by equating the
coefficients of
similar equations for the lower
2
2
(dx
dy
8^, medium.
d fdv JT- (-Tdx \dy
d (du j-*\-j-
dy \dx
dw\) + -Tr;
dzj)'
dz'\dx
Sv,, $w,,
dw
+ ~idz
we
1
dy)) get three
To the six general equations just obtained, we must add the conditions due to the surface of junction of the two media ; and at this surface we have first, u = u^ and consequently,
v=v
t
,
w=w
j}
(when x =
0)
,.(5);
ON THE REFLEXION AND REFRACTION OF LIGHT.
256
But the part of the equation (3) belonging and which yet remains to be satisfied, is
to this surface,
dv
do.
ff
dw
-
n
JJ
dy
j/
j;
dydz
5 73
,
(
+
-
s
and as 8^ = 8^, &c., we obtain, as ..
and these belong
.
8,, 4-
B,
+
-
before,
/du
dv
dw\
fdo
dw\
\dx
dy
dz)
\dy
dzl
to the particular value
cc
= 0.
The six particular conditions (5) and (6), belonging to the surface of junction of the two media, combined with the six general -equations before obtained, are necessary and sufficient complete determination of the motion of the two media, shall not here supposing the initial state of each given.
for the
We
attempt their general solution, but merely consider the propagation of a plan6 wave of infinite extent, accompanied by its reflected and refracted waves, as in the preceding paper on
Sound.
Let the direction of the axis of
0, which yet remains arbibe taken to of the plane of the the intersection trary, parallel incident wave with the surface of junction, and suppose the dis-
ON THE REFLEXION AND REFRACTION OF LIGHT.
257
turbance of the particles to be wholly in the direction of the axis of Zj which is the case with light polarized in the plane of
Then we have
incidence, according to Fresnel.
and supposing the disturbance the same for every point of the same front of a wave, w and w will be independent of z, and t
thus the three general equations
(4) will all
be
satisfied if
d*w
or
by making
=
Similarly in the lower
w
t
and It
7,
belonging to this medium.
now remains
these are
medium we have
to satisfy the conditions (5)
all satisfied
by
and
(6).
But
the preceding values provided
w=w
gdw = B dx
t
,
dw, '
dx
The formulae which we have obtained are quite general, and will apply to the ordinary elastic fluids by making But 0. for all the known gases, is independent of the nature of the
B=
A
B=
A =A
. If, therefore, we suppose gas, and consequently B^ t at least when we consider those phenomena only which depend merely on different states of the same medium, as is the case
with
light,
our conditions become*
w = w, | dw ~ _ dw \ t
dx
*
(when x
= 0) .................. (9).
dx]
for all known gases A is independent of the nature of the gas, extending the analogy rather too far, to assume that in the lurnini-
Though
perhaps
it is
17
ON THE KEFLEXION AND KEFKACTION OF LIGHT.
258
The
disturbance in the upper
medium which
contains the
incident and reflected wave, will be represented, as in the case of Sound, by
w =f(ax + ly + ct) + F(-ax+by + ct)
f belonging
to the incident,
c being a negative quantity.
These values evidently 2 = 72 (aa + (8), provided c
F to
the reflected plane wave, and Also in the lower medium,
satisfy the general equation (7) and 2 2 2 Z> and c 2 6 ) ; we have ), (a?
=%
+ ct)+F (by + ct) =/
of (by + c<) - aF'
(ly
+
which give
therefore only to satisfy the conditions (9),
f(by
;
(fy
+ ct) = af
+ ct),
(by
+ c^.
Taking now the differential coefficient of the first equation, and writing to abridge the characteristics of the functions only,
we
get
and therefore
^L_
f and
^
This
ferous
1-^a i
i
?L/
a~a a+a
t
_
cot
cot ^ /
cot 6
t
+ cot 0,
_
sin (O
sm
t
6)
(0,
+ 0)
'
being the angles of incidence and refraction. ratio
between the intensity of the incident and
ether the constants
A
and
B
reflected
must always be independent of the
state
of the ether, as found in different However, since this refracting substances. hypothesis greatly simplifies the equations due to the surface of junction of the two
media, and is itself the most simple that could be selected, it seemed natural first to deduce the consequences which follow from it before trying a more complicated one, and, as far as I have yet found, these consequences are in accordance with
observed
facts.
ON THE REFLEXION AND REFRACTION OF LIGHT.
259
is exactly the same as that for light polarized in the of incidence (vide Airy's Tracts, p. 356*), and which Fresnel plane supposes to be propagated by vibrations perpendicular to the
waves
plane of incidence, agreeably to what has been assumed in the foregoing process.
F
We will now limit the
generality of the functions/, and/, the motion to be similar to that of a the law of by supposing cycloidal pendulum; and if we farther suppose the angle of incidence to be increased until the refracted wave ceases to be
transmitted in the regular way, as in our former paper on Sound, the proper integral of the equation
a
w _ ==J t
~de~ will be
where
Tjr
= by + ct,
and aj
is
determined by !
(ii).
)
But one
of the conditions (9) will introduce sines and the way that it will be impossible to satisfy unless we introduce both sines and cosines into the value
other cosines , in such a
them
of Wj or,
w
a.
which amounts sin (ax
in the first
to the same, unless
+ by 4- ct + e) + /3 sin
(
we make
ax
medium, instead of
w = a sin (ax + by + d)
-{-
fi
sin
(
ax
+ by + ct},
which would have been done had the refracted wave been transmitted in the usual way, and consequently no exponential been
r We thus see the analytical reason for what is called the change of phase which takes place when* the reflexion of light becomes total.
introduced into the value of
*
iv
[Airy on the Undulatory Theory of Optics,
p. 109,
Art. 128.]
172
260
ON THE REFLEXION AND REFRACTION OF LIGHT. Substituting
now
and
(10)
in the equations (9),
(12),
proceeding precisely as for Sound,
= a cos e
we /3
get
cos et ,
= a sin e 4- /3 sin e
B = a sin e
Hence
by
cos e
= ft
e,
= - e,
and
.
t
and
a' a' ~ a - = = a' + r = tan 0. -
a
-=*
b
b
b
(11),
introducing
incidence.
sin e lt
/S
tan e
But by
t9
B=OL cos e + there results a
and
/u,
the index of refraction, and 6 the angle of
Thus,
/i
COS
and as e represents half the alteration of phase in passing from the incident to the reflected wave, we see that here also our result agrees precisely with Fresnel's for light polarized in the plane of incidence. (Vide Airy's Tracts, p. 362*.)
Let us now conceive the direction of the transverse vibrawave to be perpendicular to the direction in the case just considered and therefore that the actual motions
tions in the incident
;
of the particles are all parallel to the intersection of the plane of incidence (xy) with the front of the wave. Then, as the planes of the incident and refracted waves do not coincide,
it is
easy to
perceive that at the surface of junction there will, in this case, be a resolved part of the disturbance in the direction of the *
[Airy, ubi sup. p. 114, Art. 133.}
ON THE REFLEXION AND REFRACTION OF LIGHT. normal
and
;
therefore, besides the incident
wave, there
261 will, in
general, be an accompanying reflected and refracted wave, in which the vibrations are transverse, and another pair of accompanying reflected and refracted waves, in which the directions
of the vibrations are normal to the fronts of the waves. unless the consideration of the two latter waves
is
In
fact,
also intro-
duced, it is impossible to satisfy all the conditions at the surface of junction ; and these are as essential to the complete solution of the problem, as the general equations of motion.
The
direction of the disturbance being in plane (xy)
w=0,
and as the disturbance of every particle in the same front of a wave is the same, u and v are independent of z. Hence, the general equations (4) for the first medium become d*u
d*v
,
,
fdu
d _ ~ 9
fdu
dv\
SVd&
2
dy \dx
~dt*
where g
_ d ~ 9 g
d?
'
A ,
and 72
+
2
dv\
dy\dj/~~dx)>
dv\
d fdv
2
+ri ~dy)
=B
d fdu
5/ +7
du
dx\dx~~dj
.
These equations might be immediately employed in their present form ; but they will take a rather more simple form, by
making
v
_
L.
dx
and
i/r
L_
dy ,(13);
d6 d-Jr = -T---Tdy
__
_1
ax
being two functions of x,
y,
and
t,
By substitution, we readily see that the tions are equivalent to the system de
to
be determined.
two preceding equa-
dy"
.(U).
ON THE REFLEXION AND REFRACTION OF LIGHT.
262
In like manner,
^ dy get to determine
'
(15),
dx
'
we
medium we make
the second
if in
j
and ^r the equations
t
y
(16),
A
and as we suppose the constants media, we have
7
=
%
and
B
.
9,
For the complete determination
of the motion in question, it due to the surface
will be necessary to satisfy all the conditions of junction of the two media. But, since w
=
since u, v,
u
,
t
the same for both
v t are independent of
z,
and
the equations
w = 0, t
(5)
also,
and
(6)
become
A
+
dy)
du
.
dy c/
dv
__
dx
du
/
dy iJ
dy
dv,
dx
J
But since x = in the last equations, we may 0. them with regard to any of the independent variables except x, and thus the two latter, in consequence of the two former, will become provided x
differentiate
du dx
_ du
j
dx
9
dv
dv
dx
dx'
t
Substituting now for u, v, &c., their values (13) and (15), in the four resulting conditions relative to the surface of -Jr,
and
junction of the two media
may
be written,
ON THE REFLEXION AND REFRACTION OF LIGHT.
^
263
, j
da?
cfo
dr dx
dy
(when
or since
we may
differentiate
0?
with respect to
= 0)
,
the
first
and
fourth equations give
in like manner, the second
^
and third give 2
^/
~
^
2
'
dy*
which, in consequence of the general equations (14) and
(16),
become
Hence, the equivalent of the four conditions relative to the may be written
surface of junction
M
.
Ja?
dii iJ
dx
dy e/
d(f)
d^r
d t
d^r
dy
dx
dy
t
dx
(when# =
*
(17).
If we examine the expressions (13) and (15), we shall see that the disturbances due to ^> and y are normal to the front of (
the
wave
to
which they belong, whilst those which are due
to
-v/r,
ON THE REFLEXION AND REFRACTION OF LIFHT.
264 i|r /
If the coare transverse or wholly in the front of the wave. and did not differ greatly in magnitude, waves
efficients
B
A
propagated by both kinds of vibrations must in general exist, In this case, we should have in the as was before observed.
upper medium
=f(ax and and
for the
The
)"\
= % (-a (-a'# +
+
y
lower one
coefficients
Z>
=* and
c
J"
+
+ <*
being the same for
all
the functions
to simplify the results, since the indeterminate coefficients a'aa will allow the fronts of the waves to which they respectively
belong, to take any position that the nature of the problem may The coefficient of x in belonging to that reflected require.
F
wave, which, like the incident one, is propagated by transverse vibrations would have been determined exactly like a'a^a', as,
a, it was for the sake of simplicity however, it evidently = introduced immediately into our formulse.
By
substituting the values just given in the general equaand (16), there results
tions (14) 2
c
= (a + V) 7 = (a,* + V) %' = (a + V) f = 2
we have
2
2
2
(a/
+5
2 )
thus the position of the fronts of the reflected and re-
fracted waves. It
now remains
to satisfy the conditions
due to the surface
of junction of the two media. Substituting, therefore, the values (18) and (19) in the equations (17), we get
ON THE REFLEXION AND REFRACTION OF LIGHT. where
to abridge, the characteristics
265
only of the functions are
written.
By
means of the
values of sities
last four equations,
F "%"//"%/" in terms of/",
we
shall readily get the
and thus obtain the inten-
of the two reflected and two refracted waves, when the and do not differ greatly in magnitude, and the
coefficients
B
A
angle which the incident wave makes with the plane surface of junction
is
But
contained within certain limits. it
ductory remarks,
j^
was shewn that
-g
=
in
intro-
th,e
a very great quantity,
which may be regarded as infinite, and therefore g and g may be regarded as infinite compared with 7 and 7,. Hence, for all angles of incidence except such as are infinitely small, the waves and cease to be transmitted in the regular dependent on t
tf>
We
way.
t
shall therefore, as before, restrain the generality of
our functions by supposing the law of the motion to be similar to that of a cycloidal pendulum, and as two of the waves cease to
we must suppose
be transmitted in the regular way,
in the
upper medium
^ = a sin (ax + by + ct + e)+j3 sin ax + ly + ct + = ea (A sin f + B cos f and (
'*
)
and
one
in the lower
^snCa/c ,
where
= e-> (A
^ = ly +
to abridge
These substituted
t
+ fy + c*) + B cos
sin ^r
fl
t
I
^
J
ct.
in the general equations
(14)
and
(15),
give
= 9? (or, since
g and g
t
&
2
),
= a'=o/.
remains to substitute the values
equations (17), thus we get
+
are both infinite, b
It only
2
/
which belong
to the
(20),
(21) in the
surface of junction,
and
ON THE REFLEXION AND REFRACTION OF LIGHT.
266 I
A sin ^ + IB cos -^ + =
bA
cos
^
-2 (J[
-
2
5-4 y sin
1>B sin
sin
Z>a
^
ZJ5 y cos
^Q
act
cos
^ + B cos ^J = #/
T|T O
(I|T O
2
(^.^
sn
a sin (f
+ e) + Ij3 cos (i/r + e)
cos (^r
+ 5a
cos
+ e) +
sin
,) J
^jr
=
/3
^
,
(\lr
+6
cos ^r
),
cos
+ J?
)
-5 a sin y
V
V/
Expanding the two coefficients of cosi/r
last equations, comparing separately the and sin-^ OJ and observing that
9
we
suppose, 1
get
B + p cos ^ = a sine + ft sin e = a cos
e
.(23). yu,
a
t
y
In
like
manner the two
= A+A
t
first
equations of (22) will give
a sin e
ft
sin e^
= B + ^ + a cos e + /3 cos e,
a /?
= ^ - B + (/3 sin e - a sin e} 1 t
t
combining these with the system
;
(23), there results
.-.(24).
=
5 - 5, + |
(/9
sin
,
a sin
e)
ON THE REFLEXION AND REFRACTION OF LIGHT.
267
Again, the systems (23) and (24) readily give a
sm e =
*
.
a"' en
cos e
=
2 .
-J
f yu,
+ -J
a,
(25);
/8
cos
,
-
= !.
a
and therefore
(26).
When
the refractive power in passing from the upper to the lower medium is not very great, ^ does not differ much from 1.
Hence, sine and sine, are small, and cose, cos e, do not differ sensibly from unity; we have, therefore, as a first approximation,
sin
a.
2
cot 0.
2
~
~ a
a,
,
a
cot 6
sin 0, 2 sin
sin
cot
2
_ _ tan ~ sin 2(9 sin 2#, " sin 26
+ sin 20,
(0
tan (0
- 0,) + 0j
cot
,
which agrees with the formula
in Airy's Tracts, p. 358*, for light This result is
polarized perpendicular to the plane of reflexion.
only a near approximation: but the formula (26) gives the correct value of -5
,
or the ratio of the intensity of the reflected to the
; supposing, with all optical writers, that the inis of light properly measured by the square of the actual tensity of the luminiferous ether. molecules of the velocity
incident light
From
the rigorous value (26),
reflected light never
mum
value nearly
becomes
we
see that the intensity of the absolutely null, but attains a mini-
when *
[Airy, ubi sup. p.
no.]
ON THE REFLEXION AND REFRACTION OF LIGHT.
268
= ft -2
,
i.e.,
when
tan (0
which agrees with experiment, and (27) gives
this
minimum
,
value
is,
since
u
^_ ''
V-V-2
jj,
<*>
- = n,
f?
If
+ 0,) =
=3
,
as
when
IY
}<
the two media are air and water,
F _= 2
1
we
get
.
nearly.
It is evident from the formula (28), that the magnitude of this minimum value increases very rapidly as the index of so that for highly refracting substances, the intensity of the light reflected at the polarizing angle berefraction increases,
sensible, agreeably to what has been long since observed by experimental philosophers. Moreover, an inspec-
comes very
tion of the equations (25) will shew, that when we gradually increase the angle of incidence so as to pass through the polarizing angle, the change which takes place in the reflected wave is
not due to an alteration of the sign of the coefficient
/3,
but
change of phase in the wave, which for ordinary refractis very nearly substances ing equal to 180 ; the minimum value as so small to cause the reflected wave sensibly of /3 being to a
to disappear.
But
in strongly refracting substances
like dia-
mond, the coefficient /3 remains so large that the reflected wave does not seem to vanish, and the change of phase is conThese results of our theory appear to siderably less than 180. agree with the observations of Professor Airy. Trans. Vol. IV. p. 418, &c.) Lastly,
if
the velocity 7, of transmission of a
lower exceed 7 that in the upper medium, ficiently
wave
to
(Camb. Phil.
wave
in the
we may, by
suf-
augmenting the angle of incidence, cause the refracted disappear, and the change of phase thus produced in the
ON THE REFLEXION AND REFRACTION OF LIGHT. reflected
wave may
extremely easy
Let
the result. fore,
then
e,
=
readily be
after
what
precedes,
therefore, here, e,
As
found.
fju
=
the calculation
seems
it
also
,
and the accurate value of
269 is
sufficient to give e,
e is
et
and 6 as be-
given
by
2
2
tan0 2 *- -l) an9a 0-sec22-.0- (/u, . p*
The page
first
^+ 1
term of this expression agrees with the formula of and the second will be scarcely sensible
362, Airy's Tracts* ,
except for highly refracting substances. *
[Airy,
uU &up.
p. 114,
Art. 133.]
NOTE ON THE
MOTION OF WAVES IN CANALS
*
[From the Transactions of
the
Cambridge Philosophical
[Read February
18, 1839.]
Society, 1839.]
NOTE ON THE MOTION OF WAVES IN CANALS. IN a former communication* I have endeavoured to apply the ordinary Theory of Fluid Motion to determine the law of the propagation of waves in a rectangular canal, supposing f the depression of the actual surface of the fluid below that of equilibrium very small compared with its depth ; the depth 7 as well as the breadth
ft
length of a wave.
of the canal being small compared with the For greater generality, ft and 7 are supposed
vary very slowly as the horizontal co-ordinate x increases, due to the same compared with the rate of the variation of
to
*,
These suppositions are not always satisfied in the propagation of the tidal wave, but in many other cases of propa" Great Primary gation of what Mr Russel denominates the and his results will are found to agree very be so, Wave," they theoretical In with our deductions. fact, in my paper closely cause.
on the Motion of Waves,
it
has been shown that the height of
a wave varies as
ft***.
With stated
regard to the effect of the breadth ft, this is expressly by Mr Russel (vide Seventh Keport of the British Asso-
and the results given in the tables, p. 494, of same work, seem to agree with our formula as well as could
ciation, p. 425),
the
be expected, considering the object of the experiments there detailed.
In order
Primary
to
Wave
examine more particularly the way in which the propagated, let us resume the formulas,
is
18
NOTE ON THE MOTION OF WAVES IN CANALS.
274
where we have neglected the function
wave propagated
in the direction of
x
/,
which
relates to the
negative.
Suppose, for greater simplicity, that ft and 7 are constant, x being taken at the point where the wave com-
the origin of
mences when
t
= 0.
Then we may, without
altering in the
slightest degree the nature of our formulas, take the values, (1),
== gdt
But
for all small oscillations of a fluid, if (a, 5, c) are the
co-ordinates of
librium suppose
any ;
P in its primitive state, that of equithe co-ordinates of P at the end of the
particle
(x, y, z)
and <&=f<j)dt when (x, y, z) are changed into we have (vide Mecanigue Analytique, Tome II. p. 313),
time
,
Applying these general expressions
and
.
.
x=a
j=F(a 2
Neglecting (disturbance) -
;;;
,
to the formulae (1)
t
(a,
we
Z>,
c),
get
*Jgi)
we have
^
and consequently,
supposing for greater simplicity that the origin of the integral is
at
a
= 0.
Hence the value
of
x becomes
NOTE ON THE MOTION OF WAVES IN CANALS. Suppose a = length of the wave when t = and a. except when a is between the limits consider a point
P before the
wave has reached
275
then f (a)
;
= 0,
If therefore
we
it,
the whole volume of the fluid which would be required to fill the hollow caused by the depression below the surface of equi-
librium
when t =
Hence we get
0.
x =a+
j
7
x
being the horizontal co-ordinate of P,
before
the
wave
reaches P. Also, let x" be the value of this co-ordinate after the
wave
has passed completely over P, then I
dat, (a
-
1
V#7)
= 0,
and x" =
a.
<>o
If f were wholly negative, or the wave were elevated above the surface of equilibrium, we should only have to write
V
for F,
and thus
x
We see therefore, by
the transit of the
a
--y
,
and x"
a.
in this case, that the particles of the fluid are transferred forwards in the direc-
wave
and permanently deposited at rest in some distance from their original position, and
tion of the wave's motion,
a
new
place at
that the extent of the transference is sensibly equal throughout These waves are called by Mr Russel, positive
the whole depth.
ones, and this result agrees with his experiments, vide p. 423. If however f were positive, or the wave wholly depressed, it follows from our formula, that the transit of the fluid particles
would be in the opposite direction. The experimental investigation of those waves, called by Mr Russel, negative ones, has not yet been completed, p. 445, and the last result cannot therefore
be compared with experiment.
182
NOTE ON THE MOTION OF WAVES IN CANALS.
276
The
value
which we have obtained analytically
for the
extent over which the fluid particles are transferred, suggests a simple physical reason for the fact. For previous to the wave over any particle P, a volume of fluid transit of a positive
behind P, and equal to
V, is elevated
above the surface of equi-
During the transit, this descends within the surface of therefore force the fluid about P forward equilibrium, and must
librium.
through the space
admitting as an experimental fact, that after the transit of the wave the fluid particles always remain absolutely at rest.
Mr
Russel, p. 425, is inclined to infer from his experiments, that the velocity of the Great Primary Wave is that due to
gravity acting through a height equal to the depth of the centre of gravity of the transverse section of the channel below the surface of the fluid.
When
this section is a triangle of
which
channel (H), p. 443, the ordinary be applied with extreme facility. Fluid Motion of may Theory For if we take the lowest edge of the horizontal channel as the
one side
is
vertical,
as in
axis of x, and the axis of z vertical and directed upwards, the general equations for small oscillations in this case become
.
we
have, also, the conditions
= w
dz
()
z
v=d$=y
(when y = 0)
.
(
.................. (a),
z when ,
dy a being the angle which the inclined side of the channel makes with the vertical.
NOTE ON THE MOTION OF WAVES IN CANALS.
The
first
277
of these conditions is due to the vertical side, and
the second to the inclined one, since at these extreme limits the fluid particles must move along the sides.
Now that
from what has been shown in our memoir,
we may
just given,
by
= &+& Q/ +* 2
<
and
<^> /
upper
clear
)
..................... ( c ) t
only that
only remains to satisfy the condition due to the
surface.
Let therefore <>
= -&.< Then
be the equation of this surface.
paper
2
being two such functions of x and
now
It
it is
satisfy the equation (B) and the two conditions
the formula (A) of our
before cited gives
or neglecting (disturbance)
c being'the vertical depth of the fluid in equilibrium.
Also
at";the
upper surface
p = 0,
therefore
by continuing
to
2
neglect (disturbance)
(A) gives
(when z Hence, by eliminating
f,
we
= c).
get
which by (c) becomes, when we neglect terms of the order y* and z* compared with those retained,
Or
eliminating <,
by means
of
((7),
278
NOTE ON THE MOTION OF WAVES IN CANALS.
The
particular integral of proceeds in the direction of
which belonging
x
positive
to the
*-/(-'/?)
gives
A/ ~-
that
.
.;;
and hence the velocity of propagation of the wave
Mr Russel
wave
is
is
as the velocity, but at the
same time
remarks, that in consequence of the attraction of the sides of the canal fixing a portion of the fluid in its lower angle, we ought in
employing any formula
to calculate for
an
effective
depth in
Instead of adopting this method, place of the real one, p. 442. let us the formula (D) given by the common Theory of compare
Fluid Motion, with Mr Russel's experiments. And as in our theory we have considered those waves only in which the elevation above the surface of equilibrium is very small compared with the depth c, it will be necessary to select those waves in
which this condition is nearly satisfied. I have therefore taken from the Table, p. 443, all the waves in which
and have supposed g = 32J
feet
:
the results are given below.
NOTE ON THE MOTION OF WAVES IN CANALS.
279
A more perfect agreement with theory than this could scarcely be expected.
Had
the formula
would have been much
,
/-fp
=v
been used, the errors
greater.
The theory of the motion of waves in a deep sea, taking the most simple case, in which the oscillations follow the law of the cycloidal pendulum, and considering the depth as infinite, is extremely easy, and may be thus exhibited. Take
the plane (xz) perpendicular to the ridge of one of the waves supposed to extend indefinitely in the direction of the axis y, and let the velocities of the fluid particles be independent of the co-ordinate y. Then if we conceive the axis z to be directed vertically downwards, and the plane (xy) to coincide
with the surface of the sea in equilibrium,
p
we have
generally,
d
dx*
The
dz*
condition due to the upper surface, found as before,
y -rdz
--~ d?
is
.
From what precedes, it will be clear that we have now only to satisfy the second of the general equations in conjunction with This
the condition just given. niently
may be
effected
most conve-
by taking ?f x
$ = He
9 sin
A<
(v't
x),
the general equation is immediately satisfied, and the condition due to the surface gives
by which
2-7T
where X
is
,
2
evidently the length of a wave.
Hence, the velocity
of these waves varies as Vx, agreeably to what Newton asserts. But the velocity assigned by the correct theory exceeds Newton's
value in the ratio VTT to \/2, or of 5 to 4 nearly.
NOTE ON THE MOTION OP WAVES IN CANALS.
280
What
immediately precedes is not given as new, but merely on account of the extreme simplicity of the analysis employed. shall, moreover, be able thence to deduce a singular conse-
We
quence which has not before been noticed, that I
Let
when
(a, I, c)
am aware
be the co-ordinates of any particle Then, since
of.
P of the fluid
in equilibrium.
6 = jffif^' sin
?
('*-*);
A<
'=--H\ and the general formulse
(2)
-^ 27T, ^os-(^-a), give
d = a --Hre x=*a+-jda v z
^ *
.
sin
27T
.
, -(vt ^
\
H -^A cos 27T,, = c + d$> -7- = c + (v ac
v
t
a),
a).
A.
Hence,
and
any particle P revolves continually in a which the radius is
therefore
orbit, of
circular
round the point which
it would occupy in a state of equilibrium. radius of this circle, and consequently the agitation of the fluid particles, decreases very rapidly as the depth c increases,
The
and much more rapidly
for short
than long waves, agreeably to
observation. is such, that in the in the direction of of the moves circle the point upper part the motion of the wave. Hence, as in the propagation of the Great Primary Wave, the actual motion of the fluid particles is
Moreover, the direction of the rotation
P
direct
where the surface of the fluid rises above that of equiand retrograde in the contrary case.
librium,
SUPPLEMENT TO A MEMOIK
ON THE REFLEXION AND REFRACTION OF LIGHT.
From
the Transactions of the Cambridge Philosophical Society, 1839.
[Read
May
6,
1839].
SUPPLEMENT TO A MEMOIR ON THE REFLEXION AND REFRACTION OF LIGHT. IN a paper which the Society did me the honour to publish some time ago*, I endeavoured to determine the laws of Reflexion and Refraction of a plane wave at the surface of separa-
two elastic media, supposing this surface perfectly plane, and both media to terminate there abruptly neglecting also all
tion of
:
extraneous forces, whether due to the action of the solid particles of transparent bodies on the elastic medium, which is supposed to pervade their interstices, or to extraneous pressures. I am inclined to think that in the case of non-crystallized bodies
the latter cause would not alter the slightest degree
;
form
of our results in the
and possibly there would be some
difficulty in
submitting the effects of the former to calculation. Moreover, should the radius of the sphere of sensible action of the molecular forces bear any finite ratio to \, the length of a wave of
some philosophers have supposed, in order to explain phenomena of dispersion, instead of an abrupt termination of our two media we should have a continuous though rapid
light, as
the
medium
in the immediate vicinity have here endeavoured to shew, by probable reasoning, that the effect of such a change would be to diminish greatly the quantity of light reflected at
change of
state of the ethereal
of their surface of separation.
And
I
the polarizing angle, even for highly refracting substances supposing the light polarized perpendicular to the plane of inciThe same reasoning would go to prove that in this case dence. :
the quantity of the reflected light would depend greatly on minute changes in the state of the reflecting surface. I have
on the present occasion merely noticed, but not insisted upon, these inferences, feeling persuaded that in researches like the to such consequences as are not present, little confidence is due supported by a rigorous analysis. *
Supra, p. 243.
284
ON THE REFLEXION AND REFRACTION OF LIGHT.
The
principal object of this supplement has been to put the equations due to the surface of junction of two media, and be-
longing to light polarized perpendicular to the plane of inciThe resulting expressions dencej under a more simple form.
have here been made to depend on those before given in our paper on Sound, and thus the determination of the intensities of the reflected and refracted waves becomes in every case a matter As an example of the use of the new of extreme facility. formulas, the intensities of the refracted waves have been determined for both kinds of light : the consideration of which
waves had inadvertently been omitted in a former communication.
Perhaps I may be permitted on the present occasion to though I feel great confidence in the truth of the fundamental principle on which our reasonings concerning the vibrations of elastic media have been based, the same degree state, that
is by no means extended have been introduced which suppositions
of confidence
to those
for the
adventitious
sake of sim-
plifying the analysis.
Let us here resume the equations of the paper before mentioned, namely,
dx d(D
dy (fair
dx ud)
dy d"*!/* I
dy
dx
dy
dx (when
aj
= 0)
where u and v, the disturbances in the upper medium the axes x and y, are given by
_^> +,ty " dx
_ ~~~
dy
'
(17),
parallel to
ON THE REFLEXION AND REFRACTION OF LIGHT. u and t
by
v,
medium being expressed
the disturbances in the lower
similar formulae in
The two
<^>
/
and
285
ty r
last equations of (17) give, since
A&
7 = -q = -,
%
9,
and <, being accented for a moment to distinguish between the particular values belonging to the plane (yz) and their more general values <
= 6*' The
and
<
=
correctness of these values will be evident on referring to
the Memoir, formulae (20), (21), and recollecting that 7
o
Hence the
first
=a =a
'
'
. t
equation gives, since
j^ +1)(#)
;
=
^_^ = _ dy
dy
'
ind
x
,^ 1) ^.
(/
And
since
we may
may
'
dy
cf)'--2
^(^ +l)
dy Also the second equation
0,
%*
be written,
differentiate or integrate the
equations get for the conditions requisite to complete the determination of -^ and (17)
relative
to
any variable except
a?,
we
^
2
Or neglecting
-I)
2
the term which
refracting substances,
^ 2 >|r Mwhena; = 0) ..... y
is
y ,
(29).
insensible except for highly
ON THE KEFLEXION AND REFRACTION OF LIGHT.
286
___ dx dx
_
* where
= JJL
is
.(30),
.
the index of refraction.
These equations belong
to light polarized in a plane perpen-
and as and <, are insensible the surface of junction of the two from distances at sensible in the immediate vicinity of this surmedia, we have, except dicular to that of incidence,
face,
u ==
dty i
y
I
(31).
"
dx
When
light is polarized in the plane of incidence, the conditions at the surface of junction have been shewn to be
10=10,
dw
i
dw
f-
t
dx
dx
(when
a?
= 0)
(32).
J
we may differentiate or integrate of the any independent variables except a?, we see that the expressions (30) and (32) are reduced to a form equivaSince in these conditions
relative to
*
Though these equations have been obtained on the supposition that the vibrations of the media follow the law of the cycloidal pendulum, yet as (b) has disappeared, they are equally applicable for all plane waves whatever. In
fact, instead of
using the value \fft
= a, sin (a/B + by + ct),
and corresponding values of the other
quantities,
we might have taken
the infinite
series \p,
= Sa
y
sin
n (ax + by+ ct),
where a and n may have any series of values at the equivalent of an arbitrary function of 0,05
will.
But the
last expression is
+ by + ct.
Or the same equations might have been immediately obtained from (17), without introducing this consideration. The method in the text has been employed for the sake of the intermediate result (29), of
which we
shall afterwards
make
use.
ON THE EEFLEXION AND REFRACTION OF LIGHT.
287
marked (A) in our paper on Sound and the in and w the we same, general equations ^ being may imlent
that
to
;
mediately obtain the
waves, by
intensity of the reflected or refracted merely writing in the simple formulae contained in
that paper,
A=
and A,
1
=
for light polarized in
1
dence or
A=-
and A,
2
=
for light polarized perpendicular
^
/
the plane of inci-
;
to the
//
plane of incidence.
As an
example,
we
will here deduce the intensity of the
refracted w^ave for both kinds of light.
Eepresenting, therefore, the parts of w and w due to the disturbances in the Incident Reflected and Refracted waves by t
f(ax + by
+ ct), F(- ax + ly + ct), and resuming the
respectively,
paper on Sound,
first
get for
/ (a x + ly + ct) t
of our expressions (7) in the
viz.
a
2
we
and
light polarized in the
plane of incidence, where
A = A, = 1, 2
f_ t
*
2
a 1
a
,
"*"
cot
which agrees with the value given in Airy's
For
light polarized perpendicular to the plane of incidence
we have
A=
parts of
A/T
we
Tracts, p. 356*.
and A, =
2
^
and
.
If,
therefore,
we
here represent the
due to the same disturbances by /, .Fand
get
// /'
2
~
^sin0
2
7
y
cos0
cot^ cot 6
sin
2 '
!
cos 6,
cos 6 sin 6 cos 6 t sin
[* Airy, u&t sup. p. 109.]
t
f
lt
ON THE REFLEXION AND REFRACTION OF LIGHT.
288
D be the disturbance of the incident wave in its own D the like disturbance in the refracted wave, we have
Also, if plane, and
by
first
t
equation of (31),
Dsm6 = u=-^ = bf (ax and
D
t
sin 0,
= w, = *-* = jyv
(
ax
ay
retaining in ty the part due to the incident
Thus by writing
D
/
_
wave
only.
the characteristics merely,
sin
6 f]
_
2
cos
cos0
which agrees with the formula p.
in use.
(Vide Airy's Tracts,
358*.)
In our preceding paper, the two media have been supposed to terminate abruptly at their surface of junction, which would not be true of the luminiferous ether, unless the radius of the
sphere of sensible action of the molecular forces was exceedingly small compared with A,, the length of a wave of light.
In order, therefore, to form an estimate of the effect which would be produced by a continuous though rapid change of
medium in the immediate vicinity of the we will resume the conditions (29), which
state of the ethereal
surface of junction,
belong to light polarized in a plane perpendicular to that of Reflexion, viz.
and instead of supposing the index of refraction to change sudto //,, we will conceive it to pass through the denly from [* Airy, ubi sup. p.
110J
ON THE REFLEXION AND REFRACTION OF LIGHT.
289
regular series of gradations,
T being the
Then
common clear
it is
thickness of each of these successive media.
we should have
to
replace
the last system
by
and
-
(33),
and
But it is evident from the form of the equations on the right side of system (33), that the total effect due to the last terms of their second members will be far less when n is great, than that due the
corresponding term in the second equation of system we reject these second terms, and conceive
to the
therefore,
If,
(29)*.
common
terms
interval r so small that the result
due
to the first
very sensibly from that which would be a single refraction, we should have to replace by
may
produced
not
differ
the system (29) by (30), and the intensity of the reflected wave would then agree with the law assigned by Fresnel. In
however highly refracting any substance may be, homogeneous light will always be completely polarized and Sir David Brewster states at a certain angle of incidence
virtue of this law,
;
*
In
factors (29'),
fact, in 2
(/-li
-
the system (33) each of the last terms will, in consequence of the
2 /i
and as
),
&c. be quantities of the order
their
number
is
compared with the
last
term of
only n, their joint effect will be a quantity of the
order - compared with that of the term just mentioned.
19
290
ON THE REFLEXION AND REFRACTION OF LIGHT.
But the the case with diamond at the proper angle. Professor him to observed Airy appears by entirely phenomena
that this
is
inconsistent with this result (Vide Camb. Phil. Trans., Vol. IV.
what immediately precedes seems to render it probable ; that considerable differences in this respect may be due to slight changes in the reflecting surface.
p. 423)
ON THE
PROPAGATION OF LIGHT IN CRYSTALLIZED MEDIA*.
From
the Transactions of the Cambridge Philosophical Society, 1839.
[Read
May
2O> 1839,}
192
ON THE PROPAGATION OF LIGHT IN CRYSTALLIZED MEDIA. IN a former paper* I endeavoured a plane wave would be modified non-crystallized medium to another
on
this principle:
material system forces
to determine in
when
what way
transmitted from one
founding the investigation In whatever manner the elements of any
may
;
upon each
act
other, if all the internal
be multiplied by the elements of their respective directions,
the total sums for any assigned portion of the mass will always differential of some function. This principle re-
be the exact
quires a slight limitation, and
when
the necessary limitation
is
introduced, appears to possess very great generality. I shall here endeavour to apply the same principle to crystallized bodies,
and
shall likewise introduce the consideration of the effects of
extraneous pressures, which had been omitted in the former communication. Our problem thus becomes very complicated, as the function due to the internal forces, even when there are no extraneous pressures, contains twenty-one coefficients. But
with these pressures
we
are obliged to introduce six additional
some limitation, it appears quite to thence deduce hopeless any consequences which could have the least chance of a physical application. The absolute necesof some introducing sity arbitrary restrictions, and the desire coefficients;
that their to
so that without
number should be
examine how
as small as possible, induced me would be limited by confining
far our function
ourselves to the consideration of those media only in which the directions of the transverse vibrations shall always be accurately in the front of the wave. This fundamental principle of
Theory gives fourteen relations between the twentyone constants originally entering into our function; and it seems worthy of remark, that when there are no extraneous pressures, Fresnel's
the directions of polarization and the wave-velocities given by our theory, when thus limited, are identical with those assigned
by
we
Fresnel's general construction for biaxal crystals; provided suppose the actual direction of disturbance in the particles *
Supra,
p.
243.
ON THE PROPAGATION OF LIGHT
294 of the
medium
Is
parallel to the plane of polarization, agreeably
to the supposition first
If
advanced by M. Cauchy.
we admit
the existence of extraneous pressures, it will be necessary in addition to the single restriction before noticed, to suppose that for three plane waves parallel to three orthogonal
medium, and which may be denominated principal the wave-velocities shall be the same for any two of sections, the three waves whose fronts are parallel to these sections, provided the direction of the corresponding disturbances are parallel sections of our
to the line of their
intersection.
With
this
additional
position, the directions of the actual disturbances
plane wave will propagate
itself
sup-
by which any
without subdivision, and the
wave-velocities, agree exactly with those given by Fresnel, supposing, with him, that these directions are perpendicular to the
The last, or Fresnel's hypothesis, was plane of polarization. in our former But as that paper relates merely adopted paper. to the"* intensities of the waves reflected and refracted at the surface of separation of
may depend upon
two media, and
as these intensities
physical circumstances, the consideration of
which was not introduced into our former investigations, it seems right, in the present paper, considering the actual situation of the theory of light,
when
the partial differential equations
on which the determination of the motion of the luminiferous ether depends are yet to discover, to state fairly the results of both hypotheses. It is
hoped the analysis employed on the present occasion
will be found sufficiently simple, as a method has here been given of passing immediately and without calculation from the
function due to the internal forces of our medium to the equation of an ellipsoidal surface, of which the semi-axes represent in magnitude the reciprocals of the three wave-velocities, and in direction the directions of the -three corresponding disturbances by which a wave can propagate itself in one medium without
subdivision.
This surface, which
ellipsoid of elasticity,
whose
may
be properly styled the
must not be confounded with the one
by a plane parallel to the wave's front gives the of the wave-velocities, and the corresponding direcreciprocals section
IN CRYSTALLIZED MEDIA.
295
The two surfaces have only this section a very simple application of our theory would shew that no force perpendicular to the wave's front is rejected,
tions of polarization.
in
common *, and
as in the ordinary one, but that the force in question lutely nullt.
is
abso-
Let us conceive a system composed of an immense number of particles mutually acting on each other, and moreover subThen if x, y, z jected to the influence of extraneous pressures. are the co-ordinates of any particle of this system in its primi-
under pressure
tive state, (that of equilibrium
for example), the
same particle at the end of the time t will become a?', #', z, where a?', y', z are functions of a?, y, z and t. If now we consider an element of this medium, of which the primitive form is that of a rectangular parallelopiped, whose sides are dx, dy, dz, this element in its new state will assume co-ordinates of the
the form of an oblique-angled parallelopiped, the lengths of the three edges being (dx), (dy), (dz), these edges being composed of the same particles which formed the three edges dx, dy, dz in the primitive state of the element. Then will
suppose. dx'\* )
j
Again,
fdy'\*
+{-f\dzj
let
dx_
dy dz
dy'\
M
V
dy dz
dy dz
( dy'\^
+ (d^\(fdx\^+ (dy' + V/}/dx'^ \\(TZ) (Tz \(dy) (dj) (cTy)
*
[It will be seen that this
necessarily -|-
remark
is
have another common plane
not strictly correct, as the surface
must
section.]
[Referring to the values of u, v, w given in p. 301, we see that, since the is supposed to be in the front of the wave, we have
direction of vibration
But the equal to
force perpendicular to the wave's front e
a
(aw
+bv+ cw), and
is
therefore null.]
is
a -r^-+
b
r-^
+ w-^ which }
is
ON THE PROPAGATION OF LIGHT
296
_
dx
dx dz
dz
dx dz
Tx dx dx dx dyy
dx\
or
we may
dz dz
dy dy dx dy
dx dy >y
write dx' dx dy a=&ca = -r--r-+-fdz ,
,
dy ir-
dy dz
dy
+
dz dz'
j--j-> dy dz dz'dz'
dx' dx' ,dy' dy' + dx + -f- -jp = acp = -j-jax -jrdz ax dz
,
ry'
dx dy' = aby = dx' T--dy r + -fdx dx ,
-
-j-
dz
dz' dz' dy -7-+-Jdx dy dy
,
.
Suppose now, as in a former paper, that fydxdydz is the function due to the mutual actions of the particles which comSince pose the element whose primitive volume = dxdydz. must remain the same, when the sides (dx}, (dy'}, (dz'} and the cosines
7 of the angles of the elementary oblique-angled parallelepiped remain unchanged, its most general form must be a, /3,
= function
<
or since a,
b,
and a'
we may
(a, b, c, a, /3, 7),
c are necessarily positive, also
=
write
ca,
= ac/3,
/3'
2
=/(a
,
&
2 ,
c
and y=aby,
2 ,
a',
ff, 7') .................. (1).
This expression is the equivalent of the one immediately preceding, and is here adopted for the sake of introducing greater
symmetry
into our formulae.
We
will in the first place suppose that is symmetrical with regard to three planes at right angles to each other, which
we
shall take as the co-ordinate planes.
The
condition of sym-
OF THE 1
((UJnVEESITT IN CRYSTALLIZED MEDIA.
metry with respect to the plane unchanged, when we change
X^[j
^97
to
remain
(yz), will require
x x But thus a2
,
6
2 ,
c
2
and
evidently remain unaltered; moreover
a'
'
become
7
Hence we get
Applying the
we
like reasoning to the other co-ordinate planes,
see that the ultimate result will be
................ (2).
The foregoing values are perfectly general, whatever the disturbance may be; but if we consider this disturbance as very small,
we may make
x = x + Uj
= z + w,
z u, v,
and
w
being very small functions of
first order.
Then by
substitution
+
/du\*
/dv\
~dy
\dy)
\dy)
2
we
y, z,
get
W/dwV*
a?,
1+
<0
_ '
,
,
\dy)
= dv + dw Jr du du
dv dv
dw dw
Tz
ty
TyTz*~dyTz~fy~dx
du
dw
du du
dv dv
dw dw
and
t
of the
ON THE PROPAGATION OF LIGHT
298 ,
we
du
dv
du du
dv dv
dw dw
dy
dx
dx dy
dx dy
dx dy
thus see that st s2 ss ,
,
,
a',
0', 7', are
very small quantities of
and that the general formula the preceding values would take the form
the
first order,
<j)
= function
fo, s2
,
'
sa , a',
&,
(1)
by
substituting
7'),
which may be expanded in a very convergent
series of the
form
being homogeneous functions of s lf s 2 ss a.', {?, 7', of the degrees 0, 1, 2, 3, &c. each of which is very great compared with the next following one. <
,
fa, fa, &c.
,
But fa being constant, if p = the primitive density of the element, the general formula of Dynamics will give
no extraneous pressures, the supposition that the primitive state was one of equilibrium would require fa = 0, as was observed in a former paper; but this is not the case if If there were
we
introduce the consideration of extraneous pressures.
ever, as in the first case, the terms fa,
and the preceding formula
jjjpdxdy*
may
<j!>
4
,
How-
&c. will be insensible,
be written
J
Supposing p the primitive density constant, the most general form of fa will be 1
2
A, B,
C,
D, E, and
A
2
3
F being constant quantities.
In like manner the most general form of fa will contain twenty-one coefficients. But if we first employ the more parti-
IN CRYSTALLIZED MEDIA. cular value
(2),
we
299
shall get
- 2(>
==
As + Bs + Cs
+ Is* + 2Ps s + 2 QSl s
3
3
2
Or by substituting for s^ s2 s3 a, f?, 7' their values, given by system (3), continuing to neglect quantities of the third ,
order,
we
,
get
ay
dv -3-
dx)
-r \dyj
dw\*
T fdv + L(r +-r
\dz
dy)
-y-
\dzj Tiffdu
dw\*
+M(j-+r dxj \dz
Having thus the form
dw
-J--Jdy dz
_
+ 2-J^
^-jdxdy
dz dx -j-
fdu dv\ +N U+-J-) dx) ,T
2 .
.
....(4).
\dy
of the function due to the internal
actions of the particles, we have merely to substitute it in the general formula of Dynamics, and to effect the integrations by of Lagrange. Thus, parts, agreeably to the method
300
ON THE PROPAGATION OF LIGHT
^o
^l^ v
tei
+
i
-3
s
^
^ + ^
^
0>
T 3
I
g
1^
S
fe
^
a *
IK 4H ^
O*
+
+
+
^ + ^ n,
+
O^
+
^ +
8
HI
i
+
4
5
^
^
-
+ '+
ta
s>
(O
C^ ^1
C^ -4^
bJO'S
Sl-l
5
i
4 94
i^S
+
-3
1-8
+
+
03
fei
1
^
s
J >
-^
IN CRYSTALLIZED MEDIA.
301
in our indefinitely extended medium we wish to determine the laws of propagation of plane waves, we must
now
If
take, to satisfy the last equations,
u = af (ax + by + cz
+ et), v = pf(ax + by + cz + et), w =yf(ax + by + cz + et)
;
a, b, and c being the cosines of the angles which a normal to the wave's front makes with the co-ordinate axes, a, /3, 7 con-
stant coefficients, and e the velocity of transmission of a
own
perpendicular to its
front,
and taken with a contrary
Substituting these values in the equations
wave
sign.
and making
(5),
to abridge
= ( G + A) a2 + (N+ B) b* + (M+ B' = (N+ A) a*+(H+B) b* + (L + C = (M+ A) a + (L + B) 6 + (/ +
A'
r
2 ,
C) c\
2
*
c
C)
C)
c
2 ;
F'=(N+R)ab', we
= (^'
get
+ = E'a 2
- pe*) j3 + D'y
+D'j3
+ (C"
These last equations will serve to determine three values of and three corresponding ratios of the quantities a, /3, 7; and
p hence ,
r
(B
we know
the directions of the disturbance
by which a
plane wave
will propagate itself without subdivision, and also the corresponding velocities of propagation. From the form of
the equations ellipsoid
(6),
it
= A'x + B'f + z
1 *
If
is
whose equation
we
reflect
function
(4) to
equation
(7)
that
if
we
conceive an
C'z*
+ Wyz + 2E'xz + ZF'xy* ...... (7),
on the connexion of the operations by which we pass from the
the equation
may
well known, is
(7), it will
be easy to perceive that the right side of the
always be immediately deduced from that portion of the function
ON THE PROPAGATION OF LIGHT
302
and represent
its
three semi-axes
by
r',
r",
and
r'",
the directions
of these axes will be the required directions of the disturbance, and the corresponding velocities of propagation will be given by
Fresnel supposes those vibrations of the particles of the luminiferous ether which affect the eye, to be accurately in the front of the wave.
Let us therefore investigate the relation which must exist between our
coefficients,
in order to satisfy this condition for
two out of our three waves, the remaining one in consequence being necessarily propagated by normal vibrations.
For
we may
this
remark, that the equation of a plane
parallel to the wave's front is
= ax' + ly + cz If therefore
...... (a)
we make
x=x y= z
=
y'
+ l\
z
+ c\,
}
and substitute these values in the equation
(7)
of the ellipsoid ;
restoring the values of A',
equation
(a),
We thus get
r
V,
E',
f,
disappear in consequence of the the position of the wave's front. suppose,
P=4-2Z
which
^'
C',
= H=I=fjb
and
d
',
X ought to whatever may be
the odd powers of
is
d
of the second degree, ,
and
dy
d dz
by changing
it,
v,
and
w
into x, y,
and
z.
Also
.
mt
a> *> C '
This remark will be of use to us afterwards, when we come to consider the most general form of the function due to the internal actions.
IN CRYSTALLIZED MEDIA.
In
if
fact,
we
303
.
substitute these values in the function (4),
there will result
dvY
fdw
dv\*
+ fdw' ,
)
fdu
U
dv
+
dv
^
tt
which,
when
Making 1
= A,
&J .
4
dv dw\
dy)
^&J
dw\>
dudw
fdu
dv\
~\\dy
dxl
\
dw\ z
+ dw^ '
Tz
pr
+
2
dudv\ '
= J9,
=
(7,
dy dx} reduces to the last four lines.
the same substitution in the equation
= (ax + ly + czf + (Aa* + M*+Cc*) (x + y* + z*) + L(cy- fa)* + M(az- ca) + N(bx- ay? -
we
(7),
get
}
//,
2
2
Let us in the
first
...... (8). [ J
place suppose the system free from all
extraneous pressure.
Then
-4
= 0,
5 = 0,
(7=0,
and the above equation, combined with that of a plane
parallel
to the wave's front, will give
Q
= ax + by+cz
... ...................
l = L(cy- Izf + M(az- ex)* + N(bx - ay)'
2 ,
(9),
ON THE PROPAGATION OF LIGHT
304
the equations of an infinite number of ellipses which, in general, we do not belong to the same curve surface. If, however,
cause each ellipsis to turn 90 in its own plane, the whole system will belong to an ellipsoid, as may be thus shewn: Let (xyz) be the co-ordinates of any point p in its original position, and (xyz) the co-ordinates of the point p' which would coincide
withj?
when
the ellipsis
+ by' + cz,
= xx + yy' + zz, The two
turned 90 in
from the origin
since the distance
Q = ax'
is
since p
5-
f
-
ox
ex
Then
is
the same,
Op = 90.
s!
-=
y'
az
oz
cy
Hence the
=
plane.
unaltered,
since the plane
last equations give
x
is
own
its
=w
suppose.
ay
last of the equations (9)
becomes
But a-*
+ 2/ + / = o> 2
= o>2 [(b* + a
2 )
z
2
2
{(cy
- Izf + (az - ex? + (Ix - ay)*}
2
8
z
+ (c + a y + (b* 4- c )
a
Therefore
e
and our equation
finally l
We
thus see
ellipsoid to
2 )
x*
-2
(bcyz
+ abxy + acxz]
= l,
becomes
= Lx'* + My" + NJ* .................. (10).
that if
we
which the equation centre and parallel
through its when turned 90 degrees in
conceive a section (10) belongs,
by
made
in the
a plane passing
to the wave's front, this section,
own
plane, will coincide with a similar section of the ellipsoid to which the equation (8) belongs, and which gives the directions of the disturbance that will cause its
305
IN CRYSTALLIZED MEDIA. a plane wave to propagate
itself
without subdivision, and the
The change velocity of propagation parallel to its own front. of position here made in the elliptical section, is evidently equivalent to supposing the actual disturbances of the ethereal particles to
be parallel
plane usually denominated the
to the
plane ofpolarization.
This hypothesis, at first advanced by M. Cauchy, has since been adopted by several philosophers ; and it seems worthy of remark, that if we suppose an elastic medium free from all extraneous pressure, we have merely to suppose it so constituted that two
of the wave-disturbances shall be accurately in the
wave's front, agreeably to Fresnel's fundamental hypothesis, thence to deduce his general construction for the propagation of
waves in biaxal
crystals.
that the function
<
2,
which
In
fact,
in its
we
shall afterwards prove
most general form contains
twenty-one coefficients, is, in consequence of this hypothesis, reduced to one containing only seven coefficients; and that, from this last
we
form of our function,
obtain for the directions of
the disturbance and velocities of propagation precisely the same values as given by Fresnel's construction.
The above supposes, that in a state of equilibrium every When this is part of the medium is quite free from pressure. not the case, A, B, and G will no longer vanish in the equation In the first place, conceive the plane of the wave's front (8).
then a parallel to the plane (yz) tion (8) of our ellipsoid becomes
=
1,
5
= 0, c = 0,
and the equa-
*
-f
and that of a section by a plane through
its
centre parallel to
the wave's front, will be
and hence, by what precedes, the
velocities of propagation of
our two polarized waves will be
+ N. The + M.
disturbance being parallel to the axis of y t
..................................... to
the axis of
z.
20
ON THE PROPAGATION OF LIGHT
306
Similarly, if the plane of the wave's front plane (xz), the wave- velocities are .
The
is
parallel to the
disturbance being parallel to the axis x, to the axis z.
Or
if
the plane of the wave's front
parallel to
is
(xy), the
velocities are
The
.
disturbance being parallel to x,
......................................... y.
Fresnel supposes that the wave-velocity depends on the direction of the disturbance only, and is independent of the
Instead of assuming this to be
position of the wave's front.
us merely suppose it holds good for these Then we shall have three principal waves. let
generally true,
N+A = C+L, M+A=B+L, or
we may
B+N
and
write
A L =B Thus our equation
M=CN=v
(8) becomes, since
(suppose).
a2 4- 52 + c2 =
1,
+ ly + cz)* + v (y? +3/2 + ^ + (La + Mb* + Nc*) (x* + + * + L(cy- Izf + M(az- ex) + N(lx- ay)\ 2
1 =fji (ax
)
3
2
2
2/
)
2
But the two
last lines of this
formula easily reduce to
(M+ N)x* + (N+ L) f+(L + M) z* + L {aV - (by + c*) + M {% - (ax + cz) 2
2
}
And 1
hence our
= (p +
M+
JST)
last equation
x*
2
}
becomes
+ (v+N+L) y*+ (v+L+ M ^+/x (ax+ly+cz}* )
+L {aV - (by + cz}*} +M{b*if- (ax + cz)*} + JV {cV- (ax
2
-f
%)
}
........................... (11).
In consequence of the condition which was satisfied in forming the equation (8), it is evident that two of its semi- axes
307
IN CRYSTALLIZED MEDIA.
are in a plane parallel to the wave's front, and of which the
equation
is
Q = ax + by
+ cz
..................... (12);
the same therefore will be true for the ellipsoid whose equation But the as this is only a particular case of the former.
is (11),
section of the last ellipsoid
by
the plane (12)
is
evidently given '" (
'
by
l)
'
what precedes, the two axes of this elliptical section will the two directions of disturbance which will cause a wave give to be propagated without subdivision, and the velocity of pro-
By
pagation of each wave will be inversely as the corresponding semi-axes of the section: which agrees with Fresnel's construction, supposing, as he has done, the actual direction of the disturbance of the particles of the ether plane of polarization.
is
perpendicular to the
Let us again consider the system as quite free from extraneous pressure, and take the most general value of 2 containing <
twenty-one
coefficients.
du
Then,
_
dx~^ dv
Tz
we
-
shall
dw _ =<*'
+
~fy
du dz
if to
abridge,
we make
du _
du
dy~^
ds~*'
dw
_Q + dx~~P'
du
d^
_
.,
dv
_ + dx~ %
have
w
= (f r + T? + r + 2 bo tf+ 2
>
2
)
+ 2 (aij) out + 2 2
(f>
y
^+
2 (y,) yi,
2
&c. are the twenty-one coefficients which enter Suppose now the equation to the front of a wave is
where (f ), into
(/&?)
(a
),
= ax + by -f 02.
202
ON THE PROPAGATION OF LIGHT
308
Then, by what was before observed, the right side of the
which gives the directions of disequation of the ellipsoid, turbance of the three polarized waves and their respective be had from
velocities, will a?,
y and
z
}
<
by changing
2
and
u, v,
w
into
also
;
-=-
dx
-7-
,
and
,
dy
-j-
dz
into a,
and
b,
c.
We shall thus get 1
= Ax* +
%+ 2
Cz*
+ ZDyz + 2Exz + ZFxy.
Provided
4 = C?
2
'
)
+ (')
B= (rf) W + (a
2
c
c
)
2
2
2
+
+
(
7
)
2 (
7
)
s
+2
O&y) 6c
+2
() ac + 2 (fy) ok,
aa
+2
(07) ac
+2
(973)
5
5c
+2
(177)
o&, >
^= (D c + (a
2
2
5
)
D= fef) 5c + (a
2 )
2
+ G8
fo
2 )
a
2
+2
+ (^7) a2 + (a) +
^= (ff
)
ac
8
+
OS
+
(7
)
s
)
(a/9)
(<*!)
3
6
ac
+ (07) b + (a/3)
o*
+ (a/3)
c
2
+
2
a&
+2
ac
+ (a7
+
6c
(a7) be
(fa) be
(af) c
)
2
+2
(fiS)
ac,
a6
+
0&?) a5
+ 7 f) ac, (
+ (7) a& +
(^7) ac be.
But if the directions of two of the disturbances are rigorously in the front of a wave, a plane parallel to this front passing is through the centre of the ellipsoid, and whose equation
= ax + by + cz, must contain two of the semi-axes of
this ellipsoid; and theresystem of chords perpendicular to the plane will be bisected by it; and hence we get fore a
0= (A -
C) ac + E^-a^+Fbc-Dab,
= (B - C) be + D 2 - 62 + .Fac - Eab. (c )
309
IN CRYSTALLIZED MEDIA.
we
Substituting in these the values of A, B, &c., before given, shall obtain the fourteen relations following between the
coefficients of
viz.
c/>2 ,
0=M, =(), (*?)=- 2 (f)
Hence, we
(0*)=-2(7),
(07),
W = (D =
=
=
0=(7 ?),
2 (a
+ faf) = 2
2 )
(/3<)
(
7
+ (#) = 2
readily put the function
may
= - 2 (a/3),
)
<
2
s
(7
)
+
under the follow-
ing form, 2
+ (a
2
1
(a
)
- 4*) +
+2 or
(a 7 ) (a7
restoring the values of (a ), &c., our function will
by
L=
2
08") (/S
f,
2
77,
- 4#) +
- 20*) + 2
&c.,
2
2 (
7
(7
)
(a/3)
-
(a0
and making
-
G=
become
dw\* tdu, --T dv r lj + -r + -7(JT
\dx
dzj
dy
.dvdw
p
(fdu
+
du
dw\ tdu
dy
dy
+
dw\*
dv\
~
2
+ dx
dudw du fdv~ dw +
dy dz
dx .......
dy) \dz and hence we get 1
dz \dy
dxj
for the equation of the
dx)}
(12),
corresponding ellipsoid,
= G (ax + ly + czf + L(bz- cy}* +M(az- cxf + N (ay - lx)
But
if in
equation
suppose .4=0, B
0,
(8)
and
z
and corresponding function (^4), we (7 = 0, and then refer the equation to
axes taken arbitrarily in space,
we
shall thus introduce three
ON THE PROPAGATION OF LIGHT
310
new
and evidently obtain a result equivalent to and function (12). We therefore see that the
coefficients,
equation (13)
single supposition of the wave-disturbance, being always accurately in the wave's front, leads to a result equivalent to that
given by the former process; and
we
employing the simpler method we do
are thus assured that
by
not, in the case in question,
eventually lessen the generality of our result, but merely, in effect, select the three rectangular axes, which may be called the axes of elasticity of the medium, for our co-ordinate axes.
From
it is clear that the same observation and therefore the consequences before deduced
the general form of <,
applies to
it,
possess all the requisite generality.
The same
conclusions
may be
obtained, whether
we
intro-
duce the consideration of extraneous pressures or not, by direct In fact, when these pressures vanish, and we concalculation. ceive a section of the ellipsoid whose equation is (13), made by a plane parallel to the wave's front, to turn 90 degrees in its own plane, the same reasoning by which equation (10) was before found, immediately gives, in the present case, 1
= Lx'* + My* + Nz* + ZPy'z + 2Qa?V + ZEx'y'.
. .
(14),
which all the elliptical sections and corresponding to every position of
for the equation of the surface in
in their
new
situations,
the wave's front, will be found. Lastly,
when we
introduce the consideration of extraneous
clear, from what precedes, that we shall merely pressures, have to add to the function on the right side of the equation it is
(13),
the quantity
(Ac?
+ BV +
which would
arise
Also
,
-y-
,
-y-
-=-
Cc*
+ 2Dbc + 2Eac + 2Fab)
from changing into a,
6,
c,
u, v,
and
z
(x
w
into x, y,
in that part of
<
which
and
z.
is
of
v, w, agreeably to the remark in a Afterwards, when we determine the values of A, B, &c., by the same condition which enabled us to deduce
the second degree in u,
foregoing note.
IN CRYSTALLIZED MEDIA. the system (12, the following
1),
we
311
shall have, in the place of this system,
:
which *
is
applicable to the
more general case just considered*.
Vide Professor Stokes' Keport on Double Kefraction
1862, p. 265).
(British Association,
RESEARCHES ON THE VIBRATION OF
PENDULUMS IN FLUID MEDIA".
*
From
the Transactions of the Royal Society cj Edinburgh.
[Read Dec.
16, 1833.]
RESEARCHES ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA.
PKOBABLY no department greater
difficulties
of Analytical Mechanics presents than that which treats of the motion of
and hitherto the success of mathematicians therein has been comparatively limited. In the theory of waves, as presented by MM. Poisson and Cauchy, and in that of sound, fluids
;
their success
where; and
appears to have been more complete than elseto these investigations we join the researches
if
of Laplace concerning the tides, we shall have the principal important applications hitherto made of the general equations upon which the determination of this kind of motion depends.
The same
equations will serve to resolve completely a particular case of the motion of fluids, which is capable of a useful practical application;
and as I
am
not aware that
it
has yet been
noticed, I shall endeavour, in the following paper, to consider it as briefly as possible.
In the case just alluded to, it is required to determine the circumstances of the motion of an indefinitely extended nonelastic fluid, when agitated by a solid ellipsoidal body, moving parallel to itself, according to
any given law, always supposing
the body's excursions very small, compared with its dimensions. From what ^ill be shown in the sequel, the general solution But as the of this problem may very easily be obtained. principal object of our paper is to determine the alteration produced in the motion of a pendulum by the action of the sur-
sounding medium, we have insisted more particularly on the case where the ellipsoid moves in a right line parallel to one of its
axes,
and have thence proved,
that, in order to obtain the
316
ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA.
correct time of a pendulum's vibration, it will not be sufficient to allow for the loss of weight caused by the fluid
merely
medium, but that it will likewise be requisite to conceive the a quantity proportional to density of the body augmented by the density of this fluid. The value of the quantity last named the body of the pendulum is an oblate spheroid vibrating
when
equatorial plane, has been completely determined, and, the when spheroid becomes a sphere, is precisely equal to half Hence in this last case the density of the surrounding fluid.
in
its
we
shall
have the true time of the pendulum's vibration,
if
we
suppose it to move in vacua, and then simply conceive its mass augmented by half that of an equal volume of the fluid, whilst the moving force with which it is actuated is diminished by the whole weight of the same volume of fluid.
We
will
now
proceed to consider a particular case of the
motion of a non-elastic fluid over a fixed obstacle of ellipsoidal figure, and thence endeavour to find the correction necessary to reduce the observed length of a pendulum vibrating through exceedingly small arcs in any indefinitely extended medium to its true length in vacuo, when the body of the pendulum is a solid ellipsoid. For this purpose we may remark, that the equations of the motion of a homogeneous non-elastic fluid are
.
Vide Mfa CeL Liv. in. Ch.
8,
No.
33,
where
<
is
such a func-
tion of the co-ordinates x, y, z of any particle of the fluid mass, and of the time t that the velocities of this particle in the directions of shall
and tending
to increase the co-ordinates
always be represented by
-p
,
-p and ,
x
}
y,
and z
-
respectively.
Moreover, p represents the fluid's density, p its pressure, and a function dependent upon the various forces which act upon the fluid mass.
V
ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA.
317
the fluid is supposed to move over a fixed solid will be so to satisfy the equathe principal difficulty ellipsoid, tion (2), that the particles at the surface of this solid may move
When
along this surface, which
may always
be effected by making
supposing that the origin of the co-ordinates is at the centre of the ellipsoid ; X and //, being two arbitrary quantities constant regard to the variables x, y, z and a, b, c being functions of these same variables, determined by the equations
witli
:
= -+/ in
which
To it
b*
a',
= b"+f,
>',
dx*
and
^ + |! + J=l
....(4),
c are the axes of the given ellipsoid.
df
-
'
*
= c'*+f,
prove that the expression (3) satisfies the equation (2), be remarked, that we readily get, by differentiating (3),
may
my memoir
In
tions of
c>
Px
a 3bc dx
dz*
* JL *
/ay " dy * dy\ "*"
d df
dz')
_
2cV
2b*
\dx
\dz
\dy
on the Determination of the exterior and interior Attrac-
1 Ellipsoids of Variable Densities , recently communicated to the CamEDWARD FFRENCH BROMHEAD, Baronet, Sir Philosophical Society by
bridge I have given a method by which the general integral of the partial differential
equation
_
-^
&V + d^
--'
+
d?V dtf
+
d*V dtf
+
n-s dV u
du
be expanded in a series of peculiar form, and have thus rendered the determination of these attractions a matter of comparative facility. The same method
may
(2) of the present paper has the advantage of giving an general integral, every term of which, besides satisfying this The formula (3) likewise be made to satisfy the condition (6).
applied to the equation
expansion of equation,
its
may
only an individual term of the expansion in question. But in order to render the present communication independent of every other, it was thought advisable to introduce into the test a demonstration of this particular case. is
1
[Vid. supra, p.IiSs.]
ON THE VIBEATION OF PENDULUMS IN FLUID MEDIA.
318
Moreover, by the same means, the
last of the equation (4)
gives
which values being substituted in the second member of the preceding equation, evidently cause it to vanish, and we thus perceive that the value
(3) satisfies
the partial differential equa-
tion (2).
We tities
will
X and
now endeavour fju
that the
surface of the ellipsoidal
But by
and
so to determine the constant quanparticles may move along the
fluid
body
which the equation
of
differentiation, there results
as the particles
must move along the
the last equation ought to subsist,
and dz into
dx, dy
}
and
-faz
.
is
surface,
it is
when we change
clear that
the elements
their corresponding velocities
-~
~~ ,
,
Hence, at this surface
v
==
x
dtp
if
a ~J dx
But the expression
I
TJJ
(3)
T~f9
o
z U(D
u(b ~7
dy
i
~7
c
..(o)
7
dz
gives generally
'
~ dy~~cibcdy* dz
a?bc dz
""^
''
ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA. 319 and consequently
at the surface in question,
where f=
Q-\-JL r^LjiJ^L^. fy-J*lL. tf dx~ ^^J^bc^a'tVc'dx' dy~ a*b'c dy> <
(
r
These values substituted in 7/
7/r
-f-
and J- with dz
dy
(6)
give,
_ "
d
dz
when we
0,
df
jiz
'
3
a' b'c'
replace
dz
-f
,
their values at the ellipsoidal surface,
abc which may always be one of the constants
satisfied
X and
/*,
(8), v ;>
,
by a proper determination of the other remaining entirely
arbitrary.
From what
condition to which satisfied
provided
it
precedes,
is
clear that the equation (2)
the fluid is
subject
may
and
equally well be
by making
we
determine the constant quantities therein contained
by means of the equations
v c
respectively.
=x
abc
The same may
J
abc
^abc
likewise be said of the
sum
of the
before given. follows, we However, shall consider the value (3) only, since, from the results thus
three values of
in
what
obtained similar ones relative to the cases just enumerated
be found without the Instead
now
may
least difficulty.
of supposing the solid at rest, let every part of be animated with an additional common velo-
the whole system X in the direction of the co-ordinate x. city
Then
that the equation (2) and condition to which the fluid
it is
is
clear
subject,
ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA.
320 will
remain
still
satisfied.
Moreover,
to three axes fixed in space,
we
if a/, ?/, z
shall
are
now
referred
have
X
represents the co-ordinate of the centre of the ellipsoid referred to the fixed origin, we shall have
and
if
=-
\\dt
(9).
j\dt
Adding now
to
locity, the expression
\x due
the term (3) will then
to the additional ve-
become
X^' and the velocities of any point of the fluid will be given, by and its means of the differentials of this last function. But differentials evidently vanish at an infinite distance from the solid, where /= co; and consequently, the case now under consideration is that of an indefinitely extended fluid, of which the <j>
exterior limits are at rest, whilst the parts in the vicinity of
the moving body are agitated It will
now be
by
its
motions.
requisite to determine the pressure^ at any But, by supposing this mass free from
point of the fluid mass. all
extraneous action,
F= 0,
and
if
the excursions of the solid
are always exceedingly small, compared with its dimensions, the last term of the second member of the equation (1) may
evidently be neglected, and thus
we
shall have, without sensible
error, j)
or,
by
d(p
.
substitution from the last value of
Having thus motion,
let
us
u(p
,
ascertained all the circumstances of the fluid's
now
calculate its total action
upon the moving
ON THE VIBRATION OP PENDULUMS IN FLUID MEDIA. solid.
Then
321
the pressure upon any point on its surface will be in the last expression, and is
had by making /=
df Hence we
readily get for the total pressure on the
tending to increase
body
x df
C
,
dfju
f
df
"
v representing the volume of the body, p the pressure on that side where x is positive, jt? the pressure on the opposite side, and ds an element of the principal section of the ellipsoid perpendicular to the axis of x. '
If
now we
substitute for
expression will
p
its
value given from
(8),
the last
become
ofWprTi:,
Having thus the total pressure exerted upon the moving body by the surrounding medium, it will be easy thence to determine the law of its vibrations when acted upon by an proportional to the distance of its centre from In fact, let p f be the density of the body, the point of repose. v and, consequently, p f its mass, gX' the exterior force tending
exterior force
to decrease
X
1
.
Then by the
principles of dynamics,
now, in the formula (10) we substitute drawn from (9), the last equation will become If,
for
X
its
value
ON THE YIBEATION OF PENDULUMS IN FLUID MEDIA.
322
is evidently the .same as would be obtained by supposing the vibrations to take place in vacuo, under the influence of the given exterior force, provided the density of the vibrating body
which
were increased from
-a'b'c J
a*
We
thus perceive, that besides the retardation caused by the loss of weight which the vibrating body sustains in a fluid, there is
a farther retardation due to the action of the fluid
and
itself;
same as would be produced by augmenting the density of the body in the proportion just assigned, the moving force remaining unaltered. this last is precisely the
When
the
body
is
spherical,
V
we have a
c',
and the
proportion immediately preceding becomes very simple, for it will then only be requisite to increase p, the density of the body,
by ^
,
or half the density of the fluid, in order to
have the
correction in question.
The next
case in point of simplicity
is
where a
=c
;
for
then
If a
> V,
body is an oblate spheroid vibrating in its the last quantity properly depends on the equatorial plane, circular arcs, and has for value or the
I
- arc
tan (
If, on the contrary, a < value of the same integral is
= &',
V(a *- ft-))}
"
a- (a* - ft")
-
,
or the spheroid is oblong, the
V
ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA. Another very simple case of the quantities (12)
and
< &',
if a'
2
=b
where c if a > b', becomes, is
323
f
,
for
then the
first
the same quantity becomes
PB&JF*
{arc (tan
By employing the first of the four expressions immediately preceding, we readily perceive that, when an oblate spheroid vibrates in its equatorial plane, the correction now under consideration will be effected by conceiving the density of the body augmented from tan
4J
I
\At
IX
I
UC*AA
'
c\
I
2
When very
flat,
b' is
/
from p to p t
/2
f
very small compared with
we must augment
/
Y &
I
a',
7 '2\
o
J
or the spheroid is
the density
t
-t-
p nearly
;
and we thus see that the correction in question becomes proportion as the spheroid is more oblate.
less in
In what precedes, the excursions of the body of the penare supposed very small compared with its dimensions. this were not the case, the terms of the second degree in the equation (1) would no longer be negligible, and therefore the foregoing results might thus cease to be correct. Indeed, were we to attend to the term just mentioned, no advantage would even then be obtained for the actual motion of the fluid
dulum For if
;
where the vibrations are large will differ greatly from what would be assigned by the preceding method, although this method consists in satisfying all the equations of the fluid's
212
ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA.
324
motion, and likewise the particular conditions to which
it
is
subject. It
would be encroaching too much upon the Society's time on the present occasion into an explanation of the
to enter
cause of this apparent anomaly : it will be sufficient here to have made the remark, and, at the same time to observe, that when is very small, as we have all along the preceding theory will give the proper correction supposed, to be applied to bodies vibrating in air, or other elastic fluid,
the extent of the vibrations
since the error to
which
this theory leads cannot bear a
much
greater proportion to the correction before assigned, than the pendulum's greatest velocity does to that of sound.
APPENDIX.
APPENDIX. Note
to Art. 6, p. 36.
THE may
important theorem of reciprocity, established in Art. 6, be put in a clearer light by the following demonstration,
which
due
is
and
let
to Professor
Maxwell.
B
be any two points on a closed conducting surface, a unit of positive electricity be placed at a point Q,
Let A,
within the surface, then a unit of negative electricity will be so distributed over the surface that there will be no electrical
and the potential outside it will be at any point within the potential everywhere surface, due to the electricity on the surface, is a function of
force outside the surface, zero.
P
The
the positions of P, Q, and of the form of the surface. this
Denoting
by
G
it is
required to
shew that #/>=
G*\
or that the potential at P, due to the distribution on the surface caused by a unit of positive electricity at Q, is equal to the
potential at Q, due to the distribution a unit of positive electricity at P.
Let is zero,
on the surface caused by
X be any point outside the surface.
The
potential there
hence ..................... (i), is the density and dSA the element of surface, at any of the surface, and the integration is extended over the
where pA point A
whole
surface.
Also,
by
definition,
ey
.................... (2).
APPENDIX.
328
Now and
if
we
consider a unit of positive electricity placed at P, an element dSB at B, we shall have, p B be the density on if
similarly,
points outside the surface, or on zero on the surface.
for all is
Hence, substituting in equation
is
since the potential
X be on the surface, say at A, this equation becomes
Let
and as
it,
this is the
same
as
we
(2)
we
get
shall obtain for
G
p
\ the property
proved.
Note
The
equation
(r)
to Art. 10, pp. 50, 51.
=-^-J
proved on
p. 51,
may
be ex-
Let be the centre of a sphere of pressed in words as follows. radius a, and A, B, two points each of which is the electrical image of the other with respect to the sphere (i.e. let 0, A,
B
be in the same straight line, and OA OB = a?), then, if electricity be distributed in any manner over the surface of the sphere, .
the potential at as
OB
if
is
to the potential at
B
as a is to
OA
or
BP
P
a point move in such a manner that the ratio constant (= X supposs) it will describe a sphere, and C, C' be the points in which this sphere cuts AB,
For,
to
A
to a.
is
AP
if
is
AC -_AB
A( =
AB
m
329
APPENDIX.
OCAC-\- OA = -
and
A,
OG OA
Hence
And
:
potential at
A
Hence the theorem
:
is
f-
}
1
OB OC::\:
::
:
potential at
B
::
1.
-j
-
:
:
X
:
1.
proved.
The laws of the distribution of electricity on spherical conductors have been geometrically investigated by Sir William Thomson in a series of papers published in the Cambridge and Dublin Mathematical Journal. See also Thomson and Tait's Natural Philosophy, Arts. 474, 510.
Note
to Art. 12, p. 68.
In the case of a straight line uniformly covered with electricity, the form of the equipotential surface, and the law of distribution of the electricity over the surface gated as follows.
may
be investi-
Denoting the extremities of the straight line by 8, IT, we that the attraction of the line on p' may be replaced by
know
that of a circular arc of which
SH, and has Sp
',
Hp
as its
p
is
the centre, and which touches radii. Hence the direc-
bounding
tion of the resultant attraction bisects the angle Sp H, and the are equipotential surface is a prolate spheroid of which S,
H
the
foci.
dV Again, or
by
,
is
the resultant force exerted
-j
the circular arc, and therefore
dw AT
Now
y v
$P' H)
2a
by
the straight line,
APPENDIX.
330
^f
.
dw
And by
the properties of the ellipse cos
Sp
2
.
which agrees with the
H
i
f[E.
result in the text.
Note
To
to p. 246.
prove that the equilibrium of the 4
stable, unless
medium
will
be un
A > -. We have -^
fffpdxdydefy.
And Now, that
<
as
(f>j
shewn by Green,
=
+
<
2
.
in order that equilibrium may be stable, it is necessary maximum value of $, or that be a maximum
be a
value of
$a
.
In other words, that fa should never be
positive.
But
du
dv
dw\*
dv
dw
dw du
dy
Now dv dw 4-7--jay dz
dw du
du dv
+ 4-y-7-+4^ dz dx dx dv
dy
dw\*
=-
dy
du dv
APPENDIX.
331
Hence .
4
-(
dv
BK \(du -l )(dx + Ty +
A
( A
* =
dw\*
du\*
duo
Tx)
A
It thus appears that
-
o
B
d^)
du (du
+
dv\* dv
S~
B are each
-B, o
y
dw\*
of them the
an essentially negative expression. Hence, in order may always be negative, it is necessary and sufficient
coefficient of
that that
<
B
2
should be positive, and
A > - B. 3
Note to
p.
253.
Q be the positions of two particles of a medium in equilibrium distant from each other by a small interval. Let Let P,
medium
receive a
small displacement, in consequence of which these particles assume the positions P, Q', respectively. the
Let the co-ordinates of P be
Q
P then those of
Q
r
will
#,
x+
x+u,
y+ y+
_&
dw
~
co-ordinates of
du\
~
+ Sz, z + w,
Sy, z v,
be
du
du
Hence the
z>
y,
Sx,
dw
~
dw
/
P relatively to
Q' are
du
=
*
>
suppose.
APPENDIX.
332 2
If then Sf
4- Srj*
+
2
Sf
=
a
a small given quantity,
,
/>
neglecting powers and products of -y-
+2 or
p*
/<#y
[-r V<&
=
(1
(v
+ -T-)$y &s +
_
(1
+ -J-
-J-
dzj
\dx
dyj
+ 2sJ $x* +
du\
fdw
2
*
+
%
25 2)
2
+
we
have,
-y- ...
,
^ ^ Ss&e +
fdu
2
-r-
dv\
+ -r
dxj
\dy
+ 2s 82 + 2a % Bz + 2/3 &s&c + 2780%. 2
(1
3)
hence appears that all particles which, after displacement, must lie before displacement on at a given distance from It
lie
P
,
the surface of a certain ellipsoid of which the centre is at P. Hence, in general, the force called into play by the displacement
must be a function of the six
coefficients
tion of this surface referred to
P as
if
But,
the
involved in the equa-
origin.
medium be homogeneous,
the force thus called
into play will be independent of the position of this ellipsoid, and will depend upon its form and magnitude only, that is, will
be a function of the lengths of the axes of this ellipsoid. But the reciprocals of the squares on the semi-axes are the values of
X given by the equation X/ 2 +(l + 2 5l ), >
7, 2
-Xp +
7,
A
(1
/3,
+ 2*,),
_x
,
=0,
a, 2 >
/
+(l +
2s
and are therefore expressible in terms of 4
Hence,
if
the force function be called
= 1
,
+
<
2
+
3
+
...
we have
-
C
1
+s2 +*8) 3 + Z> ( 5l +5 2 + 53)
+ E {4 5lVa -
+ /3 + 72)}, - (a + /Q + 7 )) {4 (,A + Vl +* A 2
2
(a
2
)
(
8
2
2
APPENDIX.
M, A, B,
D>
C,
E
being certain arbitrary
The
By
the
it
<
Note
The
coefficients.
= and that c/>3 may be appears that fa of with Green's result. above value 2 agrees
reasoning of the text neglected.
333
to p. 269.
intensity of the reflected light attains a
minimum
value
when
accurately
a which, by (27), gives b 02
and, for the
minimum
value of
,
a
{V + (ft - l)'}i (4 + (ft - l)f (5^ - 8/t
1
If
get
/^.
=
/^
^-V+
2
I)
- Qi" + 1)' 5)* - 2/t2 + 5)* + (^ + 1)'
(5/i
-
as
when
the two media are air and water,
as
when
the two media
,
O
=
+
+ 1)* - 2/*2 + I)*
4
If
fri,"
-
=
| ^
,
1 (jj,
.
are air
'
and
glass,
we
we
nearly. ,02
And
minimum
the
value of
~
can never be greater, even
v
/5+l
2
Again, 1
,
tan
( 6l
_ +
,
e)
=
-
^j-p ""
1
334
APPENDIX.
d
d
4
l
_
(/jb
1)
b
gives
"
-1 2
2
(- n*
2
(/i
3
-!) {(,*+ 2
-l
a
At
A 4 And
2
2 (^
2
(/,
(^ - 2fS+
^ )
+ Gy? - I) 4 (^ - 2/^ + 5)* + Oi-- 1) (^-6^+1)}
1)
+
2 6yL6
-l\i
^^-2^ + 5 t tan fe 4
+ 1)
-l)
-1
2(^-2
/-/^
-l
2
"
4
/^2+I
(-
2
+ 1)
(M
2/^
^ + 6^
2 )
2
fc
2
(0.
a ^ +l "(-
-
+ a) =
+1)
8
2
-
-
^
- I)
4
'
)
4
(- ^
- (f2
-^
- 6/, + -
4
1)
(^ 4
2
2
1)}
5)*
2
APPENDIX.
Note
335
to pp. 301, 302.
results may be otherwise obtained by the consideration of the waves be propagated by normal vibrations, the one that corresponding values of a, ft 7 must be proportional to a, b y c. thus obtain, from equations (6),
These if
We
A' a
+ F'b + E'c = Fa, + B'b + D'c = E'a + D'b + 1
a
Now
Aa that .
.
b
A, B,
replacing -f
Fb + E'c
The second and
V + (2M+
we
see
we
2
Q)c
+ Aa* + Bb* +
Cc\
members of the above equation being
third
similarly transformed,
',
A
equal to
is
a
Go* + (2JV+ R)
or
,
= Pe
D' Ef, F' by their values,
G',
.
C'c
c
see that
+ (2M+ Q) c2 = Hb + (2L + P) c* + (2AT+ E] d* = /c + (2M+ Q) a* + (2L + P) c Ga*+ (2N+ R)
b
2
2
2
2
,
which leads at once to the equations the foot of and also proves that the normal given at p. 302 , /A*- Aa? - Bb* - Cc*\l .. velocity of propagation e is equal to (for all values of a, 5, c
;
;
...
,
which,
when
,
1
,
the system is free from extraneous pressure, be-
/ u,
comes If the values of
A,
B
r
C',
,
be substituted in equations for the determination of pe
extraneous pressure
fjia*+Nb*+Mc*-pe\
-N) ab, - M) ca,
(6),
2 ,
D' E', F' in terms of y
we
/*,
L, M,
N
obtain the following equation
the system being supposed free from
:
(p-N)
ab,
fjLb*+Lc*+Na*-pe
(jA-L)
be,
- M) ca - L) be (/*
(/* 2 ,
=
0,
APPENDIX.
336
which, when evaluated, becomes 2
(a
+V+c
)
- pe*) {a (M - pe ) (N- pe*) ^_ (jr _ 2
2 2 (//,
2
^
It thus appears that the velocities of propagation of the two vibrations, whose directions are in front of the wave, are given
by
the equation
a2
T?
c*
the same equation as we should obtain for the determination of the axes of the section of the ellipsoid
by
the plane
+ by + cz =
ax agreeably to equation
;
(10).
CAMBRIDGE: PRINTED BY
c.
j.
CLAY, M.A., AT THE UNIVERSITY PRESS.
FORM NO. DD6
THE UNIVERSITY
Ofr
CALIFORNIA LIBRARY
