The Collected Mathematical Papers, Volume 1 (Cambridge Library Collection - Mathematics)

dm * C.

{mh

"

mh

hht...iritt

<-)-

rQn)

a

1

(f

18

ON CERTAIN DEFINITE INTEGRALS. Substituting this value, also f\-&—TS

[3

+ ~M—u2 +•••[ f° r /( m 2 )> i n the value of U,

and observing that m = 0 gives ^ ' = 0 0 , m = l gives £' = !;, where f is a quantity determined as before by the equation a2

, we have

„ U—

62

hh/...irina -.

or writing + % for f, d^' =rf^>,the limits of
a2

b>

which, in the particular case of n = 3, may easily be made to coincide with results. The analogous integral

is apparently not reducible to a single integral.

known

4]

19

4. ON CERTAIN EXPANSIONS, IN SERIES OF MULTIPLE SINES AND COSINES. [From the Cambridge Mathematical Journal, vol. in. (1842), pp. 162—167.] IN the following paper we shall suppose e the base of the hyperbolic system of \ . logarithms ; e a constant, such that its modulus, and also the modulus of - {1 — «/l — e2}, e V( 1) are each of them less than unity; %{e" ~ } a function of u, which, as u increases from 0 to 7r, passes continuously from the former of these values to the latter, without becoming a maximum in the interval, /{e"A'(~1)} any function of u which remains finite and continuous for values of u included between the above limits. Hence, writing and considering the quantity J -leu^~1>x'{eu'"--1)} (1-ecosu) as a function of m, for values of m or u included between the limits 0 and ir, we have i I

-

^ -

-1)} cos rm dm

,o, ()

(Poisson, Mec. torn. i. p. 650); which may also be written

J -leu>H-vtf[e»'H-v}(l-ecosu)

=

S

cos

T — cos

fVr^/J6»'<-i>}cosr / J } X {6»'M)}rf M Jo 1-ecosw

J

and if the first side of the equation be generally expansible in a series of multiple cosines of m, instead of being so in particular cases only, its expanded value will always be the one given by the second side of the preceding equation. 3—2

20

ON CERTAIN EXPANSIONS,

[4

Now, between the limits 0 and tr, the function f{eu^~v}cosrx\eul'{-1)} will always be expansible in a series of multiple cosines of u; and if by any algebraical process the function fp cos ryjp can be expanded in the form fp cos rXp= 2-Z«gp8,

(ar« = O

(5);

we have, in a convergent series, y[6«V(-i)jCOsr%{ew'1"-1i} = a0H-2S"ascossM

(6).

i

Again, putting we have

- {1 — Jl - e2} = X

(7),

66

, X ~ e" - = 1 + 2S°° V cos mi 1 r 1 — e cos u

(8).

Multiplying these two series, and effecting the integration, we obtain {^^}du= l

-

i

v. *

yj

\

/

and the second side of this equation being obviously derived from the expansion of fX cos r^X by rejecting negative powers of X and dividing by 2, the term independent of X may conveniently be represented by the notation 2/X cos r^X

(10);

where in general, if FX can be expanded in the form T\ = %_Z(A8\*), we have

[A_ = A,]

f\ = £ 4 0 + 2 " 4,V

(11), (12).

(By what has preceded, the expansion of FX in the above form is always possible in a certain sense; however, in the remainder of the present papier, FX will always be of a form to satisfy the equation F (- ) = FX, except in cases which will afterwards \xy be considered, where the condition A_s = As is unnecessary.) Hence, observing the equations (4), (9), (10), J{ • / '=t I(

°° cos rm 2 cos rvX/X

(13);

from which, assuming a system of equations analogous to (1), and representing by II the product O ^ . . . , it is easy to deduce n I \J-1

s/l-e? 1/•f e « l V(-D e«2V(-D \ e^'"1' X' {eu>li-1}} (1 - e cos «)j J l ' ''-) = 2 _ £ S _ " ...ncosrmf[(2cosr x X)/(X 1 ( X,...)

(14),

4]

IN SERIES OF MULTIPLE SINES AND COSINES.

21

where F (Xu \ j . . . ) being expansible in the form

;

^

o

,

(16),

N being the number of exponents which vanish. The equations (13) and (14) may also be written in the forms

'} = 2 . : cosrm2 c o s r x X ^ * X ^~}^X J

= %_Z2_Z...n(coSrm)ILl2 cosrx^—

*

X

.~^K

+X

^/x

(17),

6

^

^[fCK, X 2 ...)... (18).

y

As examples of these formulas, we may assume (19). Hence, putting xV^-^

+ X-e^-^A,

(20),

and observing the equation > /3i € «V(-i) x '{e«^-i)}

= i_ecoSM

(21),

the equation (17) becomes

e""-1'} = Z-Z cosrmAr{1 'j^Jf^A Thus, if

= cos-\C0SU~e 1 — e cos u

6-v

(22). (23), (24)

'

cos (0 - W) = 2_ £ -y-^— cos rm'{l - | e (X + X"1)} (i (X + X"1) - e} A, ...(25), the term corresponding to r = 0 being ^ { 2 X - 2e - e (Xa +1) + 2e!!X}, — e Again, assuming f f6«V(-i>} - — =

7 1

j

~dm

-J1-^

(l-ecos«)s

(26).

22

ON CERTAIN EXPANSIONS,

[4

and integrating the resulting equation with respect to m,

_ „, sin rm^ ^ sin rm *~~~a 0-<3- = 2 _ " Ar = m + 2S" Ar

,OD. (28),

a formula given in the fifth No. of the Mathematical Journal, and which suggested the present paper. As another example, let / { < « ' M , } = cos (fl — ) ^ y l

»

J

\7*

(C S

° » - g>

J

am (1 — e cos u)3 Then integrating with respect to m, there is a term

which it is evident, & priori, must vanish. Equating it to zero, and reducing, we obtain

_e_J 1-e2

that is

X + V* '

l - ^ e C X + X-1)

l

h

= A, + ^ (X2 +1) + — (Xs + 3X) + — (\4 + 4X2 + 3) +

(32),

a singular formula, which may be verified by substituting for X its value: we then obtain sin (6 -CT)= 2S°° S i n rm A i ( x + x"~') ~ e ^

(33) ^1

The expansions of sin k(0 — or), cos k (d — or), are in like manner given by the formulae cos k (0 — or) = % °° ArL' cos kL cos rm >> '

where, to abbreviate, we have written

Forming the analogous expressions for cos k (9' - w').

sin k (6' - w'),

(34),

4]

IN SERIES OF MUPTIPLE SINES AND COSINES.

23

substituting in cos k (0 - 0') = cos k (at - at') {cos k (0 - of) cos k {& - at1) + sin k (0 -at) sin k (0' - at')} — sin k (-at — at') {sin k (0 -at) cos k {0' — at') — sin k (0' — at') cos k (0 — at)}, and reducing the whole to multiple cosines, the final result takes the very simple form cosk{0-0')=2,_Z cos {r'm' -rm + k(at-at')} ArAVcos kL cos kU (L-~)(L'•—,

--,) ...(38). .-

Again, formulas analogous to (14), (18), may be deduced from the equation 27r

I \

+ sin(nwii + r 2 m 2 . . . )

f2ir dm

dm

cos (( r 11 m11 +2r 2 2m 2). . .J) J

J^T^ ^

Jo

- ^

Sir j

(r1m1 + r3m2...) T

- ^ ... sin (nrnj + r 2 m 2 ...)T (m,, m 2 . . . ) (39),

0

Z7T

which holds from m1 = 0 to m t = 27r, &C, but in many cases universally. In this case, writing for Y (mlt m 2...) the function l7^Tle«V(-i)^{e«¥(-D}

l-ecos«

j

/

l

6

'

6

- 1 -

and observing Vr^e"2-esinu J ~ l l < ( » ^ = ^ i ^ ^ 7 ^ = 1 + 2 S { C O S S M - V - 1 sinswlV J L 1— ecosw 1—Xe""*'"1' ' an exactly similar analysis, (except that in the expansion

(41), ^ '

r(x1; x2...) = s_:s_:...^ 8l , ga ...W'..., the supposition is not made that ASu / W t - « en"«-v I

in

S2...

— A_>u _ S2 ...), leads to the result

j (J - 1 e"1'*-1) xV*''- 1 '} (1 - e cos w)

cos ( r ^ ! + r2m2...) 2n cos ( J - ^ + sin ( r ^ + r2m2...) 2" sin (r^K + r2%2X2.. .)/(X1; X 2 ^^.. .(42),

(n) being the number of variables ul7 M 2 .... Hence also/{eMlV(""1), e"8*'1"1' ...}

(2 7 ^ 1 ^qi-i«0v.+; ...)cos(r1%1

I +sm v

...)sin(r lX]

1

s/l-e5 -*e(X- x-1 !

(27-i%': (

v1

x

K+ x- )}[ ^.^

e - |e (X --xX -1))

J

J (43).

^

24

ON CERTAIN EXPANSIONS, IN SERIES OF MULTIPLE SINES AND COSINES.

[4

By choosing for f{ew'4/{~1), e^Vi-n ...}, functions expansible without sines, or without cosines, a variety of formulas may be obtained: we may instance (>>

- x->){!-2

e(\\

fX-OJA^

•x-») ^~

aing as before. 1 /V + X- )} A', t(1t(Xf r+ ^X- X) -^e P-t.(X±vaL*g_

1

1

A.so,

1

- X V 1 — e2 — \e ((XX— X"" - 1)) \.

,

,

,

>.,„

•

A'^X'e'^^^-X-e^""""' A > .*.-?'

where

1

(46).

j l - \e( X + X-1)|}(XIAV 2 ^ ^ ( X + X-i)}(X~^g +2 A > 0 e , r w ^ \ 2 ' s/l- e ^

Again,

(47),

1

and - i ^ = f = -2 (X - X-) /1 e or, what is the same thing,

IW^x+x-^Kx+x-M^fA;

0

(49).

or, comparing with (44),

which are all obtained by applying the formula (43) to the expansion of

sin COS

and comparing with the equations (25), (33).

(0 — w),

5]

25

5. ON THE INTEESECTION OF CURVES. [From the Cambridge Mathematical Journal, vol. ill. (1843), pp. 211—213.] THE following theorem is quoted in a note of Chasles' Apercu Historique &c, Memoires de Bruxelles, torn. xi. p. 149, where M. Chasles employs it in the demonstration of Pascal's theorem: " If a curve of the third order pass through, eight of the points of intersection of two curves of the third order, it passes through the ninth point of intersection." The application in question is so elegant, that it deserves to be generally known. Consider a hexagon inscribed in a conic section. The aggregate of three alternate sides may be looked upon as forming a curve of the third order, and that of the remaining sides, a second curve of the same order. These two intersect in nine points, viz. the six angular points of the hexagon, and the three points which are the intersections of pairs of opposite sides. Suppose a curve of the third order passing through eight of these points, viz. the aggregate of the conic section passing through the angular points of the hexagon, and of the line joining two of the three intersections of pairs of opposite sides. This passes through the ninth point, by the theorem of Chasles, i.e. the three intersections of pairs of opposite sides lie in the same straight line, (since obviously the third intersection does not lie in the conic section); which is Pascal's theorem. The demonstration of the above property of curves of the third order is one of extreme simplicity. Let U = 0, V = 0, be the equations of two curves of the third order, the curve of the same order which passes through eight of their points of intersection (which may be considered as eight perfectly arbitrary points), and a ninth arbitrary point, will be perfectly determinate. Let Uo, Fo, be the values of U, V, when the coordinates of this last point are written in place of x, y. Then UV0— U0V=0, satisfies the above conditions, or it is the equation to the curve required; but it is an equation which is satisfied by all the nine points of intersection of the two curves, i.e. any curve that passes through eight of these points of intersection, passes also through the ninth. C. 4

26

ON THE INTERSECTION OF CURVES.

[5

Consider generally two curves, Um = 0, Vn = 0, of the orders m and n respectively, and a curve of the rth order (r not less than TO or n) passing through the mn points of intersection. The equation to such a curve will be of the form U = Wr_m Urn + Vr-^Vn = 0,

ur_m, vr_n, denoting two polynomes of the orders 1—TO, r — n, with all their coefficients complete. It would at first sight appear that the curve U = 0 might be made to pass through as many as {1 + 2 ... + (r —TO+ 1)} + {1 + 2 ... (r — n + 1)} — 1, arbitrary points, i.e. | ( r - m + 1) (r - m + 2) + | (r - n + 1) (r - n + 2) - 1 ; or, what is the same thing, \r {r + 3) — mn + 1 (r —TO— n+l)(r —TO- n + 2) arbitrary points, such being apparently the number of disposable constants. This is in fact the case as long as r is not greater thanTO+ n — 1 ; but when r exceeds this, there arise, between the polynomes which multiply the disposable coefficients, certain linear relations which cause them to group themselves into a smaller number of disposable quantities. Thus, if r be not less than m + n, forming different polynomes of the form xf-yP Vn [a + /S = or
5]

ON THE INTEESECTION OF CURVES.

27

of the mn points of intersection above. Such a curve passes through \r (r + 3) given points, and though the mn—$(m + n-r — l)(m + n — r—2) latter points are not perfectly arbitrary, there appears to be no reason why the relation between the positions of these points should be such as to prevent the curve from being completely determined by these conditions. But if it be so, then the curve must pass through the remaining r — l)(m + n — r—2) points of intersection, or we have the theorem " If a curve of the^ rth order (r not less than m or n, not greater than m + n - 3) pass through mn-|(m + » - r - l ) (m + n — r — 2) of the points of intersection of two curves of the mth and nth orders respectively, it passes through the remaining \ (in +'n — r — 1) (m + n — r — 2)

points of intersection."

4— 2

28

[6

6. ON THE MOTION OF EOTATION OF A SOLID BODY. [From the Cambridge Mathematical Journal, vol. in. (1843), pp. 224—232.] IN the fifth volume of Liouville's Journal; in a paper "Des lois geometriques qui regissent les ddplacemens d'un systeme solide," M. Olinde Eodrigues has given some very elegant formula? for determining the position of two sets of rectangular axes with respect to each other, employing rational functions of three quantities only. The principal object of the present paper is to apply these to the problem of the rotation of a solid body; but I shall first demonstrate the formulas in question, and some others connected with the same subject which may be useful on other occasions. Let Ax, Ay, Az; Ax/t Ayn Azo be any two sets of rectangular axes passing through the point A: x, y, z, xn yn zt, being taken for the points where these lines intersect the spherical surface described round the centre A with radius unity. Join xxn yyn zz^ by arcs of great circles, and through the central points of these describe great circles cutting them at right angles: these are easily seen to intersect in a certain point P. Let Px=f, Py=g, Pz-=h; then also Px,=f, Py, = g, Pz/ = h: and Z xPxt = Z yPy, = Z zPzn = 0 suppose, 0 being measured from xP towards yP, yP towards zP, or zP towards xP. The cosines of / , g, h, are of course connected by the equation cos 2 /+ cos2 (jr + cos2 h = l. Let a, /S, 7; a', /3', j ; a", /3", 7", represent the cosines of x,x, ytx, z,x; x,y, y,y, Z/y; xtz, y,z, zp: these quantities are to be determined as functions of/, g, h, 0. Suppose for a moment, Z yPz = x,

Z zPx = y,

Z xPy = z ;

6]

ON THE MOTION OF ROTATION OF A SOLID BODY.

29

a. = cos 2 / + sin 2 /cos 0, a! = c o s / cosg + sin/sin g cos (z — 0), a" = cos/cos h + sin/sin h cos (y + 0),

then

/3 = cosg c o s / + sin g sin/cos (z + 0), /3' = cos? g + sin2 g cos 0, j3" = cos g cos h + sin g sin A cos (x — 0), 7 = cos h c o s / + sin h sin/cos (y — 0), 7' = cos h cosg + sin h sin #cos (x + 6), 7" = cos2 h + sin2 & cos 0. Also

sin (/ sin h cos x = — cos g cos A, sin A sin/cos y = — cos h c o s / s i n / sin (7 cos z = — cos / c o s gr,

and

sin g sin A sin x = c o s / sin A sin / s i n y = cos g, s i n / sin (7 sin z = cos A.

Substituting,

a = cos 2 / + sin 2 /cos 0, a' - c o s / cos g (1 — cos 0) + cos A sin #, a" = cos/cos A (1 — cos 0) — cos ^r sin 6, 0 = cos (7 cos/ (1 — cos #) — cos h sin 0, 0' =• cos2 <jr + sin2 g cos 0, /3" = cos g cos A (1 — cos 8) + c o s / s i n 0, 7 = cos h c o s / ( 1 — cos 8) + cos g sin 0, 7' = cos h cosg (1 — cos 0) — c o s / s i n 8, 7" = cos2 h + sin2 h cos 0.

Assume X = tan ^8 c o s / /^. = tan \8 cos g,v = tan ^-0 cos A, and sec2 \8 = 1 + X2 + yu2 +1>2 = K ; then

*a = l + \ 2 - / u 2 - i ' 2 ) /e/8 = 2 ( X / i - K ) ,

«7 = 2 (i/X + yu),

K a'= 2 (\/i + p), 2

2

2

K/3' = 1 + ^ - I / - X , /BY'

= 2 (>z> - X),

« a " = 2 (i/\ - / * ) , K £ " = 2 (/«/ + X),

/B7" = 1 + J>2 - X2 - /x2;

which are the formulae required, differing only from those in Liouville, by having X, /A, v, instead of \m, \n, \p; and a, a', a"; /S, /3', / 3 " ; y, 7', y", instead of a, b, c ; a', b', c'; a", b", c". I t is to be remarked, that /3', /3", / 3 ; Y"> 7» y'> a r e deduced from a, a', a", by writing fi, v, \; v, X, /x, for X, /u., 'v. Let 1 + a + /3' + 7" = v; then «u = 4, and we have Xu = /3" - 7', X2u = l + a - / 3 ' - 7 " ,

fj,v = y- a", /i2y = l - a + / 3 ' - 7 " ,

PV = a' - y 8 , J/2U = 1 - a - / 3 ' - 7 " ,

30

ON THE MOTION OF ROTATION OF A SOLID BODY.

[6

Suppose that Aw, Ay, Az, are referred to axes Ax, AY, AT,, by the quantities I, m, n, k, analogous to X, fi, v, K, these latter axes being referred to Awt, Ayt, Azlt by the quantities ln m/t nn kr Let a, b, c; a', b', c'; a", b", c"; a,, bn c,; af, bf, cf; af, bf, cf, denote the quantities analogous to a, ft, 7 ; a.', ft', 7'; a.", ft", 7". Then we have, by spherical trigonometry, the formulae a. = a a, + b af + c a", a' = a' a, + V a,' + c' af, a" = a"a, + b"a't + c"af,

ft ft' ft"

= a b, -\-b bf + c b", =a'b, + b' bf + c'bf, = a'% + b"bf + c"bf,

7 = a ct+b c/ + c c," ; 7' = a' c, + V of + c'c," ; 7" = a"c, + b"cf + c"cf.

Then expressing a, b, c; a', V, c'; a", b", c"; a/t bn c,; af, bf, cf; terms of I, m, n; I,, mt, nn after some reductions we arrive at khtv kk, {ft" - 7 ) kkt (7 — a ' ) kkt (a' - ft")

af,

bf,

cf,

in

= 4 (1 — Ul — niml — nnff, = 4II 2 suppose, = 4 (I +1, + ^m - nmf) IT, = 4 (m+ ml + Ipn — lin,) II, = 4 (n + n, + ynp. - mnf) II;

and hence II =l — ll/ — mmt — nn/, H/j, = in + ml + ltm — lmn

UX=l+l/ + n/m— nmt, Tlv = n +n, + m,n — tnnl,

which are formulas of considerable elegance for exhibiting the combined effect of successive displacements of the axes. The following analogous ones are readily obtained : P =1 +\l + fim + vn, Pml = fi — m — \n + vl,

Plt = A, — I — vm + fin, Pnt = v — n— fd + \m :

and again, P,m =yu — ml-\- \nt — vl,,

P(n = v-n/

+ fd, — \mr

These formulae will be found useful in the integration of the equations of rotation of a solid body. Next it is required to express the quantities p, q, r, in terms of X, fi, v, where as usual

dft

, dft'

„ dft"

Differentiating the values of ft>c, ft'/c, ft"ic, multiplying by 7, 7', 7", and adding, up = 2X' (7/* - 7'X + 7") + 2 / (7X - y'fi + y"v) + 2v'(-'Y~y'v

+7'»,

6]

ON THE MOTION OF ROTATION OP A SOLID BODY.

31

where A.', //, v', denote - j - , -£, -4-. Reducing, we have dt at ctt icp=2 (X' + vp' - v'n) : from which it is easy to derive the system icp = 2 ( X'+vp- v'fjb), icq = 2 ( - vX' + fi + v'X), KV = 2( fiX'-Xfi+v' ); or, determining X', fx, v', from these equations, the equivalent system 2X,' = (1 + X2) p + (\{JL- v ) q + (v\+ fi) r,

r.

The following equation also is immediately obtained, K = K (\p +fJ-q + vr). The subsequent part of the problem requires the knowledge of the differential coefficients of p, q, r, with respect to X, fi, v; X', //, v. It will be sufficient to write down the six

>w

^^j

n- - ^ - ~r fj/\

—

2f

,

from which the others are immediately obtained. Suppose now a solid body acted on by any forces, and revolving round a fixed point. The equations of motion are dt dX'~dX = ~dX' d_ dT _dT^dV dt dfi dfi dfi' dt dv' where

dv

dv'

T = \ {Atf- + Bq* + Cr*); V = 2 [](Xdx + Ydy + Zdz)] dm;

Xdx + Ydy or if Xdx Ydy ++1Zdz is not an exact differential, -^-', -=—, -=-, are independent

symbols standing for

d*

dg

dz

dX

dX

dX

32

ON THE MOTION OP ROTATION OF A SOLID BODY.

[6

see Mecanique Analytique, Avertissement, t. I. p. V. [Ed. 3, p. VII.] : only in this latter case V stands for the disturbing function, the principal forces vanishing. Now, considering the first of the above equations dT

. dp

~AP-3k , ... whence, writing

A1S

,

dq

n

dr

*-fiL + OriL

9 =

K

2

—

, , . , , dp p, q, r, K, for ff,

J jrp

Jt ^'

D

+B

dq _ ,

dr die _ , ^ , o,.'

9 9 (AP

' "

VB<

t

+

^

~K^

+

K^ ^

k

° -

td dT

dT\

and hence i ^ , - ^ j = i {Ap' - vBq' + fiGr') - - Bqv' + - Crp' + - (Ap> + Bf + Cr*) -~t(Ap-

vBq + pCr).

Substituting for X', //, v, K, after all reductions,

and, forming the analogous quantities in fi, v, and substituting in the equations of motion, these become dV {Ap + (G-B) qr)-v{Bq' f (A - 0) rp) + p [Or' + {B-A)pq} = i « ^ , dV {Or' + (B - A)pq} = £ * - ^ ,

v {Ap' + (G-B)

qr) + {Bq' + (A-G)rp}-\

fi {Ap'+(G-B)

x dV qr}+\{Bq' + (A - G)rp) + {Cr' + (B - A)pq} = 1*-^ ;

i • , , , , dp or eliminating, and replacing p, q, r, by ^ ,

dq dr ... ^ - , -r-, we obtain

6]

ON THE MOTION OF ROTATION OF A SOLID BODY.

33

to which are to be joined dX

du

dt

dt

\

dv

dtj

where it will be recollected

and on the integration of these six equations depends the complete determination of the motion. If we neglect the terms depending on V, the first three equations may be integrated in the form /~1

p> = pf 2t =

A

T)

-—<£,

q> = qi*

ri

A

T>

ra+ri5

5— 4>,

^— <j),

$

C-B ,\( . A-O~,\r.

B-A

and considering p, q, r as functions of 0, given by these equations, the three latter ones take the form K _ dX dfi dv 4qr d(ji d(j> d' K

_

d\ d4+

d/j, . dv dj>+ dj>'

_

dX

dfi

dv

=

^dtf)"

dij>+

d$]

4^ K

v

of which, as is well known, the equations following, equivalent to two independent equations, are integrals, Kg = Apil+M-ff-

fi)

v2) + 2Bq (X/J,-v)

+2Bq(/j,v+\)

+ 2 O (v\ + /*),

+ Cr

where g, g', g", are arbitrary constants satisfying

To obtain another integral, it is apparently necessary, as in the ordinary theory, to revert to the consideration of the invariable plane. Suppose g' = 0, g" = 0, g" =
then C.

5

34

ON THE MOTION OF ROTATION OF A SOLID BODY.

[6

We easily obtain, where Xo, ^0, v0, K0 are written for X, ft, v, K, to denote this particular supposition, K0Ap = 2 (z>0\0 - /*„) k, K0Bq = 2 (fioVf, + Xo) k,

naGr =

(1 + v? - V - Mo2) &;

whence, and from K0 — 1 -f X02 + yu/ + i/oa, «0Cr = (2 + 2J;,,2 — *0) k, we obtain (2 + 2v')k

(1 + ty8) ilj?

(l+

Hence, writing h = Apl* + Bq^ + C?\*, the equation

, ., i , reduces itselfC to

or, integrating,

4 dv0 h + kr -= -TT = ; 1 + v02 d(f> (k + Cr) pqr' 4 t a n - „„ = f ^

^

•

The integral takes rather a simpler form if p, q, be considered functions and becomes

of r,

2tan-^ = f A ± ^ C
lh

~

_ VpBq - Ap k + Cr '

Hence I, m, n, denoting arbitrary constants, the general values of X, /A, V, are given by the equations P o = 1 - l\0 - m/i0 - npa , P0X = I + \0 + mv0— n/i0, P 0 /i = m + Ho + n\0 — lv0 , Pov =n + vo+ Jfi0- mX0. In a following paper I propose to develope the formulae for the variations of the arbitrary constants plt qlt rlt I, m, n, when the terms involving V are taken into account.

6]

ON THE MOTION OF EOTATION OF A SOLID BODY.

35

Note. It may be as well to verify independently the analytical conclusion immediately dediicible from the preceding formulae, viz. if X, /J,, v, be given by the differential equations, dX dti dv KP= dt+vjt-fdt' Kq

Kr

_ dX ~~Vdt+

dfi dv dt + X~dt'

_

dfi

~

fi

dX

dt~xli+

dv

d~f

where K= 1 +X2 + /J?+ v2, and p, q, r, are any functions of t. Then if Xo, fi0, v0, be particular values of X, fi, v, and I, m, n, arbitrary constants, the general integrals are given by the system Po = 1 —lX0 — m.fi(, — nv0, P o \ = I +X0 + mv0 — nfi0, = m+ fio+ riX0 - lva , = n + v0 + lfi0 —

Assuming these equations, we deduce the equivalent system, ( 1 + XX0 + flfl0 + vv0) I = X — Xo + VefJ, — Vfl0, ( 1 + XX0 + fi/j,,) + w0) TO = fj, — fio + X0v — Xi'o, (1 + XX0 + yu,/i0 + vp0) n = v — v0 + /^0X — /iX 0 .

Differentiate the first of these and eliminate I, the result takes the form 0 = - (/V + V ) (X' + vfi - v'fi) - (y0 - Xofio) {-vX' + fi' + Xv') + (fi0 + Xovo) (jiX' - Xfi + v) + K0X', + (/4a + v2) (Xo + vofj,o' — vo'fio) + (y- X/M) ( - voX'o + fi0' + XoiV) - (p + Xv) (/J,OX'O - X0/i0' + v0') - KX',, where X', &c. denote -5- , &c. and K0 = 1 + Xo2 + fi02 + vf. Reducing by the differential equations in X, ft,, v\ Xo, fi0, v0, this becomes

\r(jn +Xv)} - \r (/u,0 + Xv0)} = 0 ; or substituting for X', X^, we have the identical equation \p {KOK — KK^) — 0 :

and similarly may the remaining equations be verified.

5—2

36

[7

7. ON A CLASS OF DIFFEEENTIAL EQUATIONS, AND ON THE LINES OF CURVATURE OF AN ELLIPSOID. [From the Cambridge Mathematical Journal, vol. III. (1843), pp. 264—267.] CONSIDER the primitive equation fx + gy + hz +

=0

(1),

between n variables x, y, z, the constants f, g, h being connected by the equation S(f, ff, h

)=0

(2),

H denoting a homogeneous function. Suppose that / , g, h conditions =0 M +9Vi +hzl

are determined by the (3),

=0. Then writing

X =

, z

z

n—2>

with analogous expressions for y, z to x, y, z, or eliminating/, g, h H(X,

(4),

,

•••

; the equations (3) give / , g, h, by the equation (2), Y,Z

)=0

proportional (5).

Conversely the equation (5), which contains, in appearance, n (n — 2) arbitrary constants, is equivalent to the system (1), (2). And if if be a rational integral function of the order r, the first side of the equation (5) is the product of r factors, each of them of the form given by the system (1), (2).

7]

ON A CLASS OP DIFFERENTIAL EQUATIONS, &C.

37

Now from the equation (1), we have the system fx fdx

+ gy + gdy

+ hz 4 hdx

=0 =0,

fdn~2x + gdn~2y + hdn~n-z or writing

X=

y dy

(6),

= 0,

,z , ... \ , dz , ...

(7),

dn~2y, dn~2z, ... with analogous expressions for T, Z ; then from the equations (6),/, g, h proportional to X, Y, Z : or, eliminating by (2), H(X,

Y, Z

)=0

are (8).

Conversely the integral of the equation (8) of the order (n — 2) is given either by the system of equations (1), (2), in which it is evident that the number of arbitrary constants is reduced to (n — 2); or, by the equation (5), which contains in appearance n (n — 2) arbitrary constants, but which we have seen is equivalent in reality to the system (1), (2). Thus, with three variables, the integral of H (yds — zdy, zdx — xdz, xdy — ydx) = 0

(9)

may be expressed in the form H{yzx-yxz, and also in the form where

zxx- z&, xy1-x1y)=Q

(10),

fx+gy + hz=0

(11),

H (/, g, h) = 0

(12).

The above principles afford an elegant mode of integrating the differential equation for the lines of curvature of an ellipsoid. The equation in question is (b2 - c'2) xdydz + (c2 - a2) ydzdx + (a2 - &2) zdxdy = 0

(13),

where x, y, z are connected by

siting

J = M , | = V, | = w

(15),

these become (b' - c2) udvdw + (c2 - a2) vdwdu + (a2 - b2) wdudv = 0

(16),

u+v + w = l

(17).

38

ON A CLASS OF DIFFERENTIAL EQUATIONS, AND ON THE

[7

Multiplying by — \(vdu — udv) (wdv — vdw) (udw — wdu)}-1, the first of these becomes — a?du (vdu — udv)(udw — wdu)

— b*dv (wdv — vdw) (ydu — udv)

— &div _ (udw — ivdu) (wdv — vdw)

"

but writing (17) and its derived equations under the form u+

(v+ w)=l

(19),

du + (dv + div) = 0, we deduce

— du (v + w) + u (dv + dw) = — du

(20),

i.e.

—du = — (ydu - udv) 4 (udw — wdu)

(21),

and similarly — dv = — (wdv — vdw) + (vdu — udv), — dw = — (udw — wdu) + wdv — vdw).

Substituting,

j ^ ^udw — wdu

^

(22);

=0

(28)>

/

wdv — vdw

vdu — udv

the integral of which may be written in the form J

^

L +

WV-y — VWX

e-a^

+J^L

UWy — WUX

VUX — VXU

where, on account of (17),

and also in the form

l

(24);

fu + gv + hw = 0

(25),

where / , g, h are connected by

ft^ f

c^a^ 9

9

this last equation is satisfied identically by

f

7,2

«2

°

c

n2

n2

(97)

°?L

Restoring x, y, z, xlt yx, zx for u, v, w, ux, vlt wlt the equations to a line of curvature passing through a given point xlt yx, zx, on the ellipsoid, are the equation (14) and (6s_c2)

a2 {yiz2 - yW)

( c s_ a s)

(ffl._6.)

b2 (zyW - zW)

_

c2 ( ^ y - x2y,2)

^v" ''

or again, under a known form, they are the equation (14) and

(& 2 -c 2 )^

tf-a?

f

tf-K

*_

B * C * a* + C * A * ¥ + A * B ? d>~

[

>'

7]

LINES OP CURVATURE OP AN ELLIPSOID.

39

From the equations (14), (29) it is easy to prove the well-known form

in fact, multiplying (29) by m, and adding to (14), we have the equation (30), if the equations 1

62-c2

1.

c2 - a2 1 _

1

I

a2-62 1

1

—h in

1

^- — =

1

are satisfied. But on reduction, these take the form (B2 - (?) 6 + (b2 - c2) m6 + ma? (62 - c2) = 0, 2

1

2

2

2

(32),

2

(C - A ) 9 + (c - a ) md + mb* (c - a ) = 0, (^42 -B*)e + (a2 - 62) md + m& (a2 - .&2) = 0, and since, by adding, an identical equation is obtained, m and 6 may be determined to satisfy these equations. The values of 6, m are

^kr» m

_ bV (B* - O2) + cV {(? - A2) + a262 (A* - B2) ~ (a 2 - 62) (6 2 - c2) (c2 - *•)

( 3 3 )(

}-

40

[8

8. ON LAGRANGE'S THEOREM. [From the Cambridge Mathematical Journal, vol. in. (1843), pp. 283—286.] THE value given by Lagrange's theorem for the expansion of any function of the quantity x, determined by the equation x = u + hfx

(1),

admits of being expressed in rather a remarkable symbolical form. The a priori deduction of this, independently of any expansion, presents some difficulties; I shall therefore content myself with showing that the form in question satisfies the equations

^.JF'xfadx^.JF'xdx Fx = Fu for h = 0

(2), (3),

deduced from the equation (1), and which are sufficient to determine the expansion of Fx, considered as a function of u and h in powers of h. Consider generally the symbolical expression

Jh) m v °l v i n g m general symbols of operation relative to any of the other variables dh entering into ah. Then, if B,h be expansible in the form

Bh = t(Ahm)

(5),

it is obvious that (^

(6).

8]

ON LAGRANGE'S

THEOREM.

41

For instance, u representing a variable contained in the function

Sh, and taking

a particular form of (f> (h -~\,

From this it is easy to demonstrate d du

where H'A denotes -jy 'Sh, as usual. ah

Hence also

of which a particular case is

^{(^Y^F'ufue^A^iKJ-Y^F'ue^l J du (\duj

Also,

J

dh {\duj

(11). j

(§uf'm~1 (F'u^fu) = Fu for h = 0

^ '

( 12 )-

Hence the form in question for Fx is

Fx=(±y^-iWue>/u) from which, differentiating with respect to u, and writing F instead of F',

i-kTx'Kdu)

^""' '

(14)

-

a well-known form of Lagrange's theorem, almost equally important with the more usual one. I t is easy to deduce (13) from (14). To do this, we have only to form the equation \-hf'x~'~"\du)

^"J " " ' '

(15)>

deduced from (14) by writing Fxf'x for fx, and adding this to (14), Fx = (-j-\

M

(FueVu) -h(j-)^h

(Fuf'u e^™)

du[ eVu) c.

(16). 6

42

ON LAGRANGE'S THEOREM.

[8

In the case of several variables, if x = u + hf(x,

#1...),

x1 = u1+h1f1{x,

xl...),

&c

(17),

writing for shortness F, / , / . . .

for F(u,

«,...),

/ ( u , u,...),

/ i ( « , «,-..). •••

t h e n the formula is

{where / ' ( * ) is written to denote -j- f(x, or the coefficient of AnV>

in the expansion of F(x,

i8

=

T

xx ...), &c.)

Xt

)

(19)

^ V ^

1.2 ... w. 1.2 ... From the formula (18), a formula may be deduced for the expansion of F(x, xx ), in the same way as (13) was deduced from (14), but the result is not expressible in a simple form by this method. An apparently simple form has indeed been given for this expansion by Laplace, Mecanique Celeste, [Ed. 1, 1798] torn. I. p. 176; but the expression there given for the general term, requires first that certain differentiations should be performed, and then that certain changes should be made in the result, quantities z, z , are to be changed into zn, z^ ; in other words, the general term is not really expressed by known symbols of operation only. The formula (18) is probably known, but I have not met with it anywhere.

43

9. DEMONSTRATION OF PASCAL'S THEOREM. [From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 18—20.] LEMMA 1. Let U = Ax + By + Cz = Q be the equation to a plane passing through a given point taken for the origin, and consider the planes TJX = 0,

U2 = 0,

Us = 0,

Ut = 0,

Us = 0,

U6 = 0 ;

the condition which expresses that the intersections of the planes (1) and (2), (3) and (4), (5) and (6) lie in the same plane, may be written down under the form A2,

As,

At

B1,

2? 2 ,

B3,

Bt

Clt

O2,

Cs,

Ct

As,

Ait

As,

Ae

B3,

B A,

Bs,

B6

C3,

Ct,

( 7 6 , Ce

. LEMMA

= 0.

Ax,

.

2. Representing the determinants &c.

by the abbreviated notation 123, &c; the following equation is identically true: 345.126 - 346.125 + 356.124 - 456.123 = 0. 6—2

44

DEMONSTRATION

OP PASCAL'S

[9

THEOREM.

This is an immediate consequence of the equations

Xx,

x3,

xit

xs,

xe

x3,

2/s,

2/4,

2/5,

2/6

2/3, 2/4,

3s,

£4,

Z»,

Z6

*^3,

*^4,

^5,

^6

2/3,

2/4,

2/5,

2/6

2/l,

2/2,

•^I,

^2,

= 0.

xt,

Consider now the points 1, 2, 3, 4, 5, 6, the coordinates of these being respectively x1, y l t z± xe, y6, ze. I represent, for shortness, the equation to the plane passing through the origin and the points 1, 2, which may be called the plane 12, in the form

consequently the symbols 12s, 12,,, 12, denote respectively y^ — y^,

zxx2 — z2xlt

xxy2—x2yx,

and similarly for the planes 13, &c. If now the intersections of 12 and 45, 23 and 56, 34 and 61 lie in the same plane, we must have, by Lemma (1), the equation 12., 45s, 23*, 56,,,

.

12,, 45,, 23,, 56,,

.

12,, 45,, 23,, 56,,

.

.

= 0.

23*, 56^, 34*, 61* 23j, 56Z, 34,, 61, Multiplying the two sides of this equation by the two sides respectively of the equation = 612.345, 2/2,

•

x 4, 2/3,

2/4,

zt, and observing the equations = 612,

112 = 0, &c.

45

DEMONSTBATION OF ]PASCALS TB[EOB

9] this becomes

612

.

= 0,

•

645, 145, 245, 623, 123,

423, 523

.

156, 256, 356, 456, .

. 534

361, 461, 561 reducible to 612 534

= 0;

145, 245, 123, 156,

423

256,

•

356, 456 361, 461

or, omitting the factor 612 . 534 and expanding, 145 . 256 . 423.361 + 245.123. 456.361 - 245.123.356. 461 - 245.156 . 423.361 = 0. Considering for instance xe, ys, ze as variable, this equation expresses evidently that the point 6 lies in a cone of the second order having the origin for its vertex, and the equation is evidently satisfied by writing x6, y6, ze = x1} ylt zlt or x3, y3, z3, or x4, yit e4, or xs,ys, zs, and thus the cone passes through the points 1, 3, 4, 5. For xe,y6, zB=a;2, y2, za, the equation becomes, reducing and dividing by 245.123, 452". 321 - 352. 421 + 152 . 423 = 0, which is deducible from Lemma (2), by writing xe, y6, ze = #2, y2, z2, and is therefore identically true. Hence the cone passes through the point (2), and therefore the points 1, 2, 3, 4, 5, 6 lie in the same cone of the second order, which is Pascal's Theorem. I have demonstrated it in the cone, for the sake of symmetry; but by writing throughout unity instead of z, the above applies directly to the case of the theorem in the plane. The demonstration of Chasles' form of Pascal's Theorem (viz. that the anharmonic relation of the planes 61, 62, 63, 64 is the same with that of 51, 52, 53, 54), is very much simpler; but as it would require some preparatory information with reference to the analytical definition of the similarity of anharmonic relation, I must defer it to another opportunity.

46

[10

10. ON THE THEORY OF ALGEBRAIC CURVES. [From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 102—112.] SUPPOSE a curve defined by the equation U = 0, U being a rational and integral function of the mth order of the coordinates x, y. It may always be assumed, without Ios3 of generality, that the terms involving xm, ym, both of them appear in U; and also that the coefficient of ym is equal to unity: for in any particular curve where this was not the case, by transforming the axes, and dividing the new equation by the coefficient of ym, the conditions in question would become satisfied. Let Hm denote the terms of U, which are of the order m, and let y — ax, y — fix ...y — \x be the factors of Hm. If the quantities a, fi ...X are all of them different, the curve is said to have a number of asymptotic directions equal to the degree of its equation. Such curves only will be considered in the present paper, the consideration of the far more complicated theory of those curves, the number of whose asymptotic directions is less than the degree of their equation, being entirely rejected. Assuming, then, that the factors of Hm are all of them different, we may deduce from the equation C = 0 , by known methods, the series

y = ax + a.'+ — +

y =Xx + X -{

,

(1).

1-

and these being obtained, we have, identically, U = (y - ax -a' - ...) (y - fix- fi' - . . . ) . . . (y -Xx

-X'

- ...)

(2),

10]

ON THE THEORY OP ALGEBRAIC CURVES.

47

the negative powers of x on the second side, in point of fact, destroying each other. Supposing in general that fx containing positive and negative powers of x, Efx denotes the function which is obtained by the rejection of the negative powers, we may write (8),

«x...

the symbol E being necessary in the present case, because, when the series are continued only to the power x~m+1, the negative powers no longer destroy each other. We may henceforward consider U as originally given by the equation (3), the m ( m + l ) quantities a, a'...a<m>, /3, fi'... /3<m), ...X, X'...X(m> satisfying the equations obtained from the supposition that it is possible to determine the following terms a(m+i)) ygc»+i) . . . \ C H - D , ... so that the terms containing negative powers of x, on the second side of equation (2), vanish. I t is easily seen that a, /3 ... X, a', / 3 ' . . . X' are entirely arbitrary, a", fi"...X" satisfy a single equation involving only the preceding quantities, a.'", j3'"... X'" two equations involving the quantities which precede them, and so on, until a(m), yS(m) ... X(m), which satisfy (m—1) relations involving the preceding quantities. Thus the m ( m + l ) quantities in question satisfy \m(m — 1) equations, or they may be considered as functions of m (m + 1) — \m (m — 1) = \m (m + 3) arbitrary constants. Hence the value of U, given by the equation (3), is the most general expression for a function of the m th order. I t is to be remarked also that the quantities a(m+1), /S(m+1) ... X(m+1), are all of them completely determinable as functions of a, /3 ... X,... a<m>, /3<m> ... X « . The advantage of the above mode of expressing the function U, is the facility obtained by means of it for the elimination of the variable y from the equation U—0, and any other analogous one V = 0. In fact, suppose V expressed in the same manner as U, or by the equation -Bx...- — ) ...(y-Kx...-^-)

(4),

n being the degree of the function V. I t is almost unnecessary to remark, that A, B ...K,...A are to be considered as functions of £n(re + 3) arbitrary constants, and that the subsequent Ain+1), B(n+1> ...Kin+1) ... can be completely determined as functions of these. Determining the values of y from the equation (3), viz. the values given by the equations (2); substituting these successively in the equation (5), analogous to (2), and taking the product of the quantities so obtained, also observing that this product must be independent of negative powers of x, the result of the elimination may be written down under the form E

f | ( Z ) +

... (6),

48

ON THE THEORY OF ALGEBRAIC CURVES.

[10

the series in { } being continued only to x~mn+1, because the terms after this point produce in the whole product only terms involving negative powers of x. It is for the same reason that the series in ( ), in the equations (3) and (4), are only continued to the terms involving x~~m+1, #~"+1 respectively. The first side of the equation (6) is of the order mn, in x, as it ought to be. But it is easy to see, from the form of the expression, in what case the order of the first side reduces itself to a number less than mn. Thus, if n be not greater than m, and the following equations be satisfied, A=a,

A®=a® ...A«-v = **-»,

r$>n

(7),

1

B = ft B® =/3< » . K=K,

R® = «w ... KP-v = KC-1*,

V > u,

the degree of the equation (6) is evidently mn- r — s... - v, or the curves U= 0, F = 0 intersect in this number only of points. If mn — r — s... — v = 0, the curves JJ=O and V=0 do not intersect at all, and if mn — r—s — v be negative, = —w suppose, the equation (6) is satisfied identically; or the functions U, V have a common factor, the number a> expressing the degree of this factor in x, y. Supposing the function V given arbitrarily, it may be required to determine U, so that the curves U=0, V= 0 intersect in a number mn—k points. This may in general be done, and done in a variety of ways, for any value of k from unity to \m (m + 3). I shall not discuss the question generally at present, nor examine into the meaning of the quantity mn — | m ( m + 3) {— \m (2m — m — 3)} becoming negative, but confine myself to the simple case of U and V, both of them functions of the second order. It is required, then, to find the equations of all those curves of the second order which intersect a given curve of the second order in a number of points less than four. Assume in general

then A", B" satisfy A" + B" = 0, and putting B" = .KD, and therefore A" = - - 7 ^ - 5 , A —H -A — li and reducing, we obtain

V=(y - Ax - A')(y -BxSimilarly assume

U= E (y — ax — a'

B') + K. ) (y — fix — /3' — — ) ,

then a", /3" satisfy a" + /3" = 0, and putting fi" = ~~3, reducing, we obtain

V = (y - ax - a') (ff-fix- ff) + k.

and therefore a" = - ——=, and

10]

ON THE THEORY OF ALGEBEAIC CURVES.

49

Suppose (1) U=0, V= 0 intersect in three points, we must have a = A, or the curve U=0 must have one of its asymptotes parallel to one of the asymptotes of V= 0. (2) The curves intersect in two points. We must have a = A, a' = A', or else a = A, fi = B; i.e. JJ=O must have one of its asymptotes coincident with one of the asymptotes of the curve F = 0, or else it must have its two asymptotes parallel to those of V = 0. The latter case is that of similar and similarly situated curves. (3) Suppose the curves intersect in a single point only. a' = A', <x" = A", which it is easy to see gives

Then either a = A,

or else a = A, a! = A', /3 = B, which is the case of one of the asymptotes of the curve U=0, coinciding with one of those of the curve V=0, and the remaining asymptotes parallel. As for the first case, if a, ax are the transverse axes, 6, 6X the inclinations of the two asymptotes to each other, these four quantities are connected by the equation a? _ tang cos2 \Q ~a? ~ tan 6X cos2 \Q^'

and besides, one of the asymptotes of the first curve is coincident with one of the asymptotes of the second. This is a remarkable case; it may be as well to verify that £7~=0, V=0 do intersect in a single point only. Multiplying the first equation by y — Bx — B', the second by y —fix—fi',and subtracting, the result is (A-P)(y-Bx-B')-(A-B)(y-0x-/3') = O, reducible to

Combining this with V = 0, we have an equation of the form y — Bx — D = 0. from this and y — Ax — C = 0, x, y are determined by means of a simple equation.

And

(4) Lastly, when the curves do not intersect at all. Here a = A, a! = A', /3 = B, /3' = £', or the asymptotes of U=0 coincide with those ef F = O ; i.e. the curves are similar, similarly situated, and concentric: or else a = A, a' = A', a" = A", /3 = B; here U ={y - Ax - A'){y -Bx- $') +K, or the required curve has one of its asymptotes coincident with one of those of the proposed curve; the remaining two asymptotes are parallel, and the magnitudes of the curves are equal. In general, if two curves of the orders m and n, respectively, are such that r asymptotes of the first are parallel to as many of the second, s out of these asymptotes

c.

7

50

ON THE THEORY OF ALGEBRAIC CURVES.

[10

being coincident in the two curves, the number of points of intersection is ran — r—s; but the converse of this theorem is not true. In a former paper, On the Intersection of Curves, [5], I investigated the number of arbitrary constants in the equation of a curve of a given order p subjected to pass through the mn points of intersection of two curves of the orders m and n respectively. The reasoning there employed is not applicable to the case where the two curves intersect in a number of points less than mn. In fact, it was assumed that, W = 0 being the equation of the required curve, W was of the form uU+vV; u, v being polynomials of the degrees p — m, p — n respectively. This is, in point of fact, true in the case there considered, viz. that in which the two curves intersect in mn points; but where the number of points of intersection is less than this, u, v may be assumed polynomials of an order higher than p — m, p — n, and yet uTJ+vV reduce itself to the order p. The preceding investigations enable us to resolve the question for every possible case. Considering then the functions U, V determined as before by the equations (3), (4), suppose, in the first place, we have a system of equations a = A, Assume

$ =B

0 = H (t equations)

(8).

P = (y — ax — ...){y — fix — ...) ... (y — 6x — ...), Q ={y -Ax-.

..){y-Bx-...)...(y-Hx

T =(y -ix-...)

whence

... {y-KX

U = PT,

-...);

-...),

V=QV.

Suppose . U-ET.

V= = EV.P(ET + AT) -ET.Q (EV + = ET. EV . (P - Q) + EV . P. AT - # T . Q. = E [ET. EV. (P - Q) + EV. PAT - E? . Q. = II suppose.

In this expression ET, EW are of the degrees m — t, n — t, AT, A ^ of the degree - 1, and P, Q, P — Q of the degrees t, t, t - \ respectively. The terms of II are therefore of the degrees m + n — t — 1, m — 1, n — 1 respectively, and the largest of these is in general m + n — t — 1. Suppose, however, that m + n — t — 1 is equal to m — 1 (it cannot be inferior to it), then t = n; *P becomes equal to unity, or A'*' vanishes. The remaining two terms of II are ET (P — Q), PAT, which are of the degrees m - 1, n—1 respectively. II is still of the degree m—1, supposing m >n. If m = n, the term PAT vanishes. II is still of the degree m — 1. Hence in every case the degree of II is m + n — t — 1: assuming always that P — Q does not reduce itself to a degree lower than t — 1, (which is always the case as long as the equations

10]

ON THE THEORY OF ALGEBRAIC CURVES.

51

a' = A', ft'= B' 6' = H' are not all of them satisfied simultaneously). It will be seen presently that we shall gain in symmetry by wording the theorem thus: the degree of II is equal to the greatest of the two quantities m + n — t — 1, m — 1. Suppose next, in addition to the equations (8), we have

a' = A',

0' = B'

t,'=F,

t' equations, t$>t... (8').

Then, taking T', V, P', Q' the analogous quantities to T, M*, P, Q, and putting

EW.U-ET.

V=W,

we have, as before,

IT = E {EV. EW. (P' - Q') + EW. P'AV - ET. The degree of which case the m + n — if — 2, greatest of the the case where

P'- Q' is t'-2 degree may be n — 1, m — 1. quantities m+n t' = n — 1 would

(unless simultaneously a" = A", /3" = B" £' = F\ in lower). The degrees, therefore, of the terms of II' are Or we may say that the degree of II' is equal to the — t' — 2, m — 1 ; though to establish this proposition in require some additional considerations.

Continuing in this manner until we come to the quantity II1*"11, the degree of this quantity is the greatest of the two numbers m + n — t{k~1] — k, m — 1. And we are satisfied, so that the series may suppose that none of the equations a{k) = A[k) II, II' II i*"1* is not to be continued beyond this point. Considering now the equation of the curve passing through the mn — t — t'... — t{k~1} points of intersection of 27=0, V— 0, we may write

W = uU+vV+pU+p'W

+p*-« n*-1>=0

for the required equation; the dimensions of u, v, p, p' ...pik"1) p — m, p—n;

p-m —n + t+1 or p — m+1; p — m—n + t' + 2 or p—m + 1;

(9),

being respectively

p — m — n + t{k~l) +k or p — m + 1,

the lowest of the two numbers being taken for the dimensions of p, p' ...p{k~1}. Also, if any of these numbers become negative, the corresponding term is to be rejected. In saying that the degrees of p, p' p(k-v have these actual values, it is supposed 1 that the degrees of II, II' II*" actually ascend to the greatest of the values p — m — n + t+1,

or m — 1 ;

m + n — t' — 2, or m — 1 ;

— m + n — <*-1) +k, or m — 1 .

The cases of exception to this are when several of the consecutive numbers t, t'...t{k~1} are equal. In this case the corresponding terms of the series II, II' II*-1*, are also equal. Suppose for instance t, t' were equal, II, II' would also be equal. A term of p of an order higher by unity than p — m —n + t + 1, or p-m+1, which is the highest term admissible, produces in pYl a term identical with one of the terms of p'U; so that nothing is gained in generality by admitting such terms into p. The equation (9), with the preceding values for the dimensions of p, p' ^*~1), may be 7—2

52

ON THE THEORY OF ALGEBRAIC CURVES.

[10

employed, therefore, even when several consecutive terms of the series t, t' t^~i> are equal. It will be convenient also to assume that p — m is not negative, or at least for greater simplicity to examine this case in the first place. u, U, and v, V, contain terms of the form af-y^JJ, aPy*V, a + $ ^> p — m, p —n; pH contains terms of this form, and in addition terms for which a + /3 = p + l — m, y + S = p + l—n. It is useless to repeat the former terms, so that we may assume for p, a homogeneous function of the order p — m — n + t + 1, or p — m+1; in which case pH consists only of terms for which a + f3=p+l — m, j + 8 = p + l—n. And the general expression of p contains p — m—n + t + 2, or p — m+ 2, arbitrary constants. Similarly p'U.' contains terms of the form of those in uU, vV, pH, and also terms for which ot+/3=p + 2 — m, y+8 = p+2—n; the latter terms only need be considered, or p' may be assumed to be a homogeneous function of the order p — m — n + t' + 2, or p — m + 1, containing therefore p — m— n —1'+3, or p — m + 2 arbitrary constants. Similarly p*"^ Assume

contains p — m — n + t{k~1} + k + 1 or p — m+ 2 arbitrary constants.

(p-m-n + t + 2\

fp-m-n + t'+8\

(p-m-n + t*-^ + k+l

where, in forming the value of V the least of the two quantities in ( ) is to be taken; this value also, if negative, being replaced by zero. The number of arbitrary constants in p, p' ...p1*"11 is consequently equal to V. The numbers of arbitrary constants in u, v, are respectively {1+2 i.e.

+(p-m

+ l)} and {1+2

+ (p-n+l)}

i 0 - m + 1) (p - m + 2), and \ (p - n + 1) (p - n + 2);

thus the whole number of arbitrary constants in W, diminished by unity (since nothing is gained in generality, by leaving the coefficient (for instance of yf) indeterminate, instead of supposing it equal to unity) becomes | (p - m + 1) (p - m + 2) + \ (p -n + 1) (p - n + 2) + V - 1, reducible to Jp (p + 3) + \{p — m - n + 1) (f) — m — n + 2) — mn + V. By the reasonings contained in the paper already referred to, if p -f k — m — w + 1 be positive, to find the number of really disposable constants in W, we must subtract from this number a number ^(p + k— m — n+l)(p + k — m — n + 2). Hence, calling the number of disposable constants in W, we have

4> = i/3 0 + 3 ) + i (P ~ m - n +l) (fi - m - n + 2 ) - mn + v - A where

iip + k — m — n+1

A = 0, if p + k—ra — n + 1 be negative or zero

be positive; and y is given by the equation (9).

(12),

10]

ON THE THEORY OF ALGEBRAIC CURVES.

53

Also, if 0 be the number of points through which the curve W = 0 can be drawn, including the points of intersection of the curves U=0, F = 0 , then 0=(f> + (mn-t-t'

-i*- 1 )) or

0=hp(p + 3) + ?(p-™>-n + l)(p-'m-n + 2)+V-A~t-t'...-t*-v

(13).

Any particular cases may be deduced with the greatest facility from these general formulae. Thus, supposing the curves to intersect in the complete number of points mn, we have 8 being zero or unity according as p < m + n — 1 or p > m + n — 1. Reducing, we have, for p $> m + n — 3,

0 = \p (p + 3) + J (p — m — n +1) (p — m — n + 2) — mn, and for p > m + n — 3,

4> = ip (P + 3 ) ~ mn> Suppose, in the next place, the curves have t parallel pairs of asymptotes, none of these pairs being coincident. Then p$> m + n — t— 2, $ = |JO (p + 3) + \ (p — m — n + 1) (p — m — n + 2)— mn, p>m + n — t — 2, p 1f> m +n — 2, 0 = \p (p + 3) + £ (p - m - n + 2) (p - m - n + 3), p >m + n — 2, <j) = \p (p + 3) — mn +1, In these formulae, if t be equal to 2 or greater than 2, the limiting conditions are more conveniently written p^m

+ n — t—2;

p $> m + n— t— 2>m + n — 4 ;

p>m + n — 4>.

Similarly may the solution of the question be explicitly obtained when the curves have t asymptotes parallel, and t' out of these coincident, but the number of separate formulae will be greater. In conclusion, I may add the following references to two memoirs on the present subject: the conclusions in one point of view are considerably less general even than those of my former paper, though much more so in another. Jacobi, Theoremata de punctis intersectionis duarum curvarum algebraicarum; Crelle's Journal, vol. xv. [1836, pp. 285—308]; Pliicker, Theoremes gendraux concernant les equations a plusieurs variables, d'un degre1 quelconque entre un nombre quelconque d'inconnues. D° vol. XVI. [1837, pp. 47—57].

54

ON THE THEORY OF ALGEBRAIC CURVES.

[10

Addition. As an exemplification of the preceding formulae, and besides as a question interesting in itself, it may be proposed to determine the asymptotic curves of the rth order of a given curve having all its asymptotic directions distinct,—r being any number less than the degree of the equation of the given curve. A curve of the rth order, which intersects a given curve of the mth o r c j e r i n a number of points, = mr — -|r (r + 3), is said to be an asymptotic curve of the rtb order to the curve in question. Suppose, as before, U = 0 being the equation to the given curve, DEFINITION.

U=E [y -ax-

} ... [y -\x

...

...

and let 6, $ ... to denote any combination of r terms out of the series a... X, and &', '... &/, &c. ... the corresponding terms out of a'... X', &c. Then, writing -*»...-_)(y-^...-^-—) \J/(m)

(where the quantities (m», ^i™-1), ? » , n'...Wm> are entirely determinate, since, by what has preceded, &, '...£l' satisfy a certain equation, 6", ",...il" two equations Q\m)t <j><m); ... il(™)(m-l) equations), we have V=0 for the required equation of the asymptotic curve. It is obvious that the whole number of asymptotic curves of the order r, is n (n — 1) ... (n — r + 1), viz. 1 . 2 ... r curves for each combination of —^

1 2

asymptotes. Some particular instances of asymptotic curves will

be found in a memoir by M. Plucker, Liouvilles Journal, vol. I. [1836, pp. 229—252], Enumeration des courbes du quatrieme ordre, &c. The general theory does not seem to be one to which much attention has been paid.

55

11]

11. CHAPTERS IN THE ANALYTICAL GEOMETRY OF (n) DIMENSIONS. [From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 119—127.] CHAP.

1. On some preliminary formulce.

I TAKE for granted all the ordinary formulae relating to • determinants. It will be convenient, however, to write down a few, relating to a certain system of determinants, which are of considerable importance in that which follows: they are all of them either known, or immediately deducible from known formulae. Consider the series of terms

•"-lj

the number of the quantities A

**-2

-"-n

K being equal to q (q < n).

Suppose q + 1

vertical rows selected, and the quantities contained in them formed into a determinant, this may be done in

-=—~

—-—~- different ways. The system of determinants

so obtained will be represented by the notation (2),

and the system of equations, obtained by equating each of these determinants to zero, by the notation ! =0.

(3).

5G

ANALYTICAL GEOMETRY OF (n) DIMENSIONS.

[11

The -^—= ", ^ — ~ equations represented by this formula reduce themselves to r J 1.2 (n-q + 1) H (n — q) independent equations. Imagine these expressed by (l) = 0, (2) = 0 (n-q)=0 (4), any one of the determinants of (2) is reducible to the form (5), where ®1( ©2...@»_3 are coefficients independent of xlt xt...xn. be replaced by

The equations (3) may =0

\x,

•(6),

and conversely from (6) we may deduce (3), unless = 0

(7).

T 2 , . . . Tn

(The number of the quantities \ , may also be expressed in the form

is of course equal to n.) The equations (3)

/J,...T

(8), ... o)9^T2,... \qAn ... + wqKn the number of the quantities X, ft ...a> being q. And conversely (3) is deducible from (8), unless = 0.

\ , Xq,

(9).

...

CHAP. 2. On the determination of linear equations in xlt x2, ... xn which are satisfied hy the values of these quantities derived from given systems of linear equations. It is required to find linear equations in xlt...xn which are satisfied by the values of these quantities derived—1. from the equations £(' = 0, W = 0 ... US' = 0; 2. from the equations S " = 0, 23" = 0 ... J i , " = 0 ; 3. from &'"=(), W = 0...W = 0, &c. &c, where npn,

...+

(1),

Bnxn,

and similarly ^ " , 23",..., ST", 23'",..., &c. are linear functions of the coordinatesxlt x2,... xn.

57

ANALYTICAL GEOMETRY OF (ft) DIMENSIONS.

11]

Also r', r"... representing the number of equations in the systems (1), (2).. . and k the number of these given systems, « - 1 or

(k-

r ' + r" +• ...

Assume

••• =

... = &c

X

(2),

the latter equations denoting the equations obtained by equating to zero the terms involving x1, those involving x2, &c.... separately. Suppose, in addition to these, a set of linear equations in V, /jf... X", fi"... so that, with the preceding ones, there is a sufficient number of equations for the elimination of these quantities. Then, performing the elimination, we thus obtain equations M* = 0, where ? is a function of a-'u # 2 ... which vanishes for the values of these quantities derived from the equations (1) or (2) ...&c. The series of equations M* = 0 may be expressed in the form

jj 8 ' , 23', ...<&', ' A,', £/,...(?/, A '

R'

h '

A^',...O'\, A "

A II

CHAP. 3.

=0 f)

II

A in

(3).

if III

On reciprocal equations.

Consider a system of equations nxn=Q,

(1),

... + Knxn = 0, (r in number). The reciprocal system with respect to a given function (U) of the second order in xx, x2...xn, is said to be dXlU ,

dX2U,...d^U

A1,

A2, ...

An

Ki,

K2,...

Kn

=0

•(2),

(n — r in number). It must first be shown that the reciprocal system to (2) is the system (1), or that the systems (1), (2) are reciprocals of each other. C. 8

58

ANALYTICAL GEOMETRY OF (il) DIMENSIONS.

Consider, in general, the system of equations ,(3).

dX2U... +andx,lU = 0 X! dXi U + X2 dx, U... + \ n dXn U = 0.

Suppose

2JJ— 1 (a2)a?a2 + 22 (a/3) xaxp, so that dXlU=%(sa)x,a

(4), (5).

The equations (3) may be written «1{«1(l2) + a 2 (12)...+a ) l (ln)j + ... + ^ l {a 1 (nl) + a2(n2)...H-afB(n2)}=0 &c.

(6),

and forming the reciprocals of these, also replacing d^U, dxJJ... by their values, we have xx (I 2 ) + x2(12) + ...xn (In), ...fr (nl) + »2 (n2) ...+«„(w a, (I2) + a2 (12) + ... an.{ln), ...a, (nl) + a2 (n2) ... + «„ >-! (I2) + X2 (12) + ... \ n (In),...

= 0..,...(7).

\ , (nl)

From these, assuming (I2), (21),

.(8)

(12),... (In) j * 0 (22),...(2«.

we obtain, for the reciprocal system of (3),

(9).

=0 , a 2 , . . . an 1;

X 2 , . . . Xm

Now, suppose the equations (3) represent the system (2) ; their number in this case must be n — r. Also if 0 represent any one of the quantities a, /3 ... X, we have n6n

=Q

(10),

By means of these equations, the system (9) may be reduced to the form nxn,

... K1x1+K2x2...+Knxn, 0

xr+1, x,+3, ... xn

r+

i,

Xr+2,

...

which are satisfied by the equations (1). Hence the reciprocal system to (2) is (1), or (1), (2) are reciprocals to each other.

11]

59

ANALYTICAL GEOMETRY OF (fl) DIMENSIONS. THEOREM.

Consider the equations

(&' =o, 33' =o...e&' =0) (8"=0, 23" =0... j i " =0),

(i2),

(a"'=o, 23"'=o...m'"=o), &c. of Chap. 2. The equations = 0, ...(13),

i*P , dxJJ

= 0, A '

A'

ff2', ... Gn'

Oi",

O "

O2'

&c. which are the reciprocals of these systems, represent taken conjointly the reciprocal of the system of equations (3) of the same chapter. Let this system, which contains n— {(n — r) + (n — r1) + ...} equations, be represented by ,.. + anasn= 0 nzn=

(14),

0.

The reciprocal system is XlU,

dxTJ,...dxJJ

= 0

(15),

«„ ?1 >

containing (n — r) + (n — r') + &c

M

>•••

5»

equations.

Also, by the formulas in Chap. 2, a1x1+

...+anxn

= \1'@L' + fi1'W+

. . . c/CBr' (X, fi...

cr, r' i n n u m b e r ) .

.(16), writing 8 = n—{(n — r) + (n—r')+ ...}. Also, a s s u m i n g a n y a r b i t r a r y q u a n t i t i e s %, i/ 2 ••• Vn ••• s e t s b e i n g ( r ' — #),) s u c h t h a t

...+Vnxn

= \ fl+1 'a

<$>$••• 4>n (the number of

a3 /' 23'+...o-/ €5'.

(17),

60

ANALYTICAL GEOMETRY OF (n) DIMENSIONS.

[11

From the equations (15) we deduce the (n — r) equations

dxJJ, dxJJ,...dxJJ

=0

(18).

Vn

Vi , , •••

n

Hence, writing

7)=\1'A+/i1'B+...
(19).

and reducing, by the formula (8) of Chap. 1, we have

dxJJ, dxJJ,...dx,JJ = 0 (?/ ,

(20);

( ? / , . . . Gn

and similarly may the remaining formulae of (13) be deduced from the equation (15). Hence the required theorem is demonstrated, a theorem which may be more clearly stated as follows :— The reciprocals of several systems of equations form together the reciprocal of the equation which is satisfied by the values of the variables which satisfy each of the original systems of equations. (The theorem requires that the number of all the reciprocal equations shall be less than the number of variables.) Conversely, consider several systems of equations, the whole number of the equations being less than the number of variables. These systems, taken conjointly, have for their reciprocal, the equation which is satisfied by the values satisfying the reciprocal system of each of the given systems. CHAP.

4.

On some properties of functions of the second order.

Suppose, as before, U denotes the general function of the second order, or (21). Also let V denote a function of the second order of the form 2-2

Pi,

(22),

P i , ••• P n

(H being the symbol of a homogeneous function of the second order, and the number r of the quantities a,ft...p, being less than n — 1). [Observe that alt &, ... plt... a n, @n,---Pn, or say the suffixed quantities a, /3,...p (r in number) are used to denote coefficients: a, /3, without suffixes, are any two numbers in the series of suffixes 1, 2, 3, ...n.] Then 2U-2kV, h arbitrary, is of the form 2 [ < K 2 + 2 2 [a/3] xjcf

(23).

11]

61

ANALYTICAL GEOMETRY OF («) DIMENSIONS.

Suppose Xlt X 2 , . . . X n determined by the equations (24),

equations that involve the condition that k satisfies an equation of the order n — r, as will be presently proved. Then shall x1 = Xl ...xn = Xn satisfy the system of equations, which is the reciprocal of =0

Pi,

(25).

Pi,---Pn

To prove these properties, in the first place we must find the form of Consider the quantities %A, %B, ... %L, (n — r) in number, of the form »*»,

(20),

L n ocn; where, if © represent any of the quantities A, B ... L, • (27),

... + 2 (AB) £, Hence, if

2 V = 2, {a-\ x£ + 2.

22 (AB) UB(28),

we have for the coefficients of this form {I2} = t (A') A,* + 22 (AB) A1B1,

{12} = 2 (A') A,A2 +

(AB) (A& + A A),

and consequently the coefficients of 2V — 2kV are [1»] = (1»)-*{P},

[12] = (12) - ft {12}.

Hence, 6 representing any of the quantities a, ^ ... p, (29),

V.

62

ANALYTICAL GEOMETBY OF (il) DIMENSIONS,

whence also

[11

61 [P ] + ... 6n [In] = 0, (V ) + ... 6n (In), e\ [ni] + . . . en [«» ] = ex (HI) +... en (n»).

Hence, the equations for determining Xlt...Xn

may be reduced to

X,[a, (I2) +...«„(In)] + X2[a, (21) ...+an (2n)] ...+Xn{a1(n\)

... + <xn (n2)] = 0 ... (30),

31 (21)... X, [Pl (1«) +... pn (In)] +X,[Pl(21)...+Pn 1, 1]+ x [n, 1] +

= 0,

(2n)]... + Xn [pn (nl) ... + ^ (n2)] = 0.

X 3 [r + 1, 2]...+

Xn[r + l,n]

=0, = 0.

Xt[n, 2]...+

Eliminating X, ...Xn, since the first r equations do not contain k, the equation in this quantity is of the order n — r. Next form the reciprocals of the equations (25). These are dxTJ, dx.TJ,...dxJJ Alt A2.... An hi ,

X/2 , . . .

(31).

=0

Jjn

From which we may deduce a^TJ ...+andXnU, 13^11...+!3ndxU,...PldXLU...pndx,U, 0 0 0 6

0

0

dXr+lU, ...dJJ Ar+lu...An

=0...(32),

Lr+l,...Ln

which are evidently satisfied by x1 = X1, a;2 = X 2 ... xn — Xn. In the case of four variables, the above investigation demonstrates the following properties of surfaces of the second order. I. If a cone intersect a surface of the second order, three different cones may be drawn through the curve of intersection, and the vertices of these lie in the plane which is the polar reciprocal of the vertex of the intersecting cone. II. If two planes intersect a surface of the second order through the curve of intersection, two cones may be drawn, and the vertices of these lie in the line which is the polar reciprocal of the line of intersection of the two planes. Both these theorems are undoubtedly known, though I am ntt able to refer for them to any given place.

12]

63

12. ON THE THEORY OF DETERMINANTS. [From the Transactions of the Cambridge Philosophical Society, vol. VIII. (1843), pp. 1—16.] THE following Memoir is composed of two separate investigations, each of them having a general reference to the Theory of Determinants, but otherwise perfectly unconnected. The name of " Determinants" or " Resultants" has been given, as is well known, to the functions which equated to zero express the result of the elimination of any number of variables from as many linear equations, without constant terms. But the same functions occur in the resolution of a system of linear equations, in the general problem of elimination between algebraic equations, and particular cases of them in algebraic geometry, in the theory of numbers, and, in short, in almost every part of mathematics. They have accordingly been a subject of very considerable attention with analysts. Occurring, apparently for the first time, in Cramer's Introduction & VAnalyse des Lignes Courbes, 1750: they are afterwards met with in a Memoir On Elimination, by Bezout, Memoires de I'Academie, 1764; in two Memoirs by Laplace and Vandermonde in the same collection, 1774; in Bezout's Thiorie ginirale des Equations algebriques [1779]; in Memoirs by Binet, Journal de VEcole Poly technique, vol. ix. [1813]; by Cauchy, ditto, vol. x. [1815]; by Jacobi, Crelle's Journal, vol. xxn. [1841]; Lebesgue, Liouville, [vol. II. 1837], &c. The Memoirs of Cauchy and Jacobi contain the greatest part of their known properties, and may be considered as constituting the general theory of the subject. In the first part of the present paper, I consider the properties of certain derivational functions of a quantity U, linear in two separate sets of variables (by the term " Derivational Function," I would propose to denote those functions, the nature of which depends upon the form of the quantity to which they refer, with respect to the variables entering into it, e.g. the differential coefficient of any quantity is a derivational function. The theory of derivational functions is apparently one that would admit of interesting developments). The particular functions of this class which are here considered, are closely connected with the theory of the reciprocal polars of surfaces of the second order, which latter is indeed a particular case of the theory of these functions. In the second part, I consider the notation and properties of certain functions resolvable into a series of determinants, but the nature of which can hardly be explained independently of the notation.

64

[12

ON THE THEORY OF DETERMINANTS.

In the first section I have denoted a determinant, by simply writing down in the form of a square the different quantities of which it is made up. This is not concise, but it is clearer than any abridged notation. The ordinary properties of determinants, I have throughout taken for granted; these may easily be learnt by referring to the Memoirs of Cauchy and Jacobi, quoted above. It may however be convenient to write down the following fundamental property, demonstrated by these authors, and by Binet.

a,

0,...

*, P,

«•',

pa + crfi...,

pa! +
p'a+
p'a' + p'ar'

an equation, particular cases of which are of very frequent occurrence, e.g. in the investigations on the forms of numbers in Gauss' Disquisitiones Arithmetica [1801], in Lagrange's Determination of the Elements of a Comet's Orbit [1780], &c. I have applied it in the Cambridge Mathematical Journal [1] to Carnot's problem, of finding the relation between the distances of five points in space, and to another geometrical problem. With respect to the notation of the second section, this is so fully explained there, as to render it unnecessary to say anything further about it at present. § 1. On the properties of certain determinants, considered as Derivational Functions. Consider the function ...)+ (1), + •••)

+

(n lines, and n terms in each line) ; and suppose =

.(2).

a,/3,,

(The single letter K being employed instead of KU, in cases where the quantity KU, rather than the functional symbol K, is being considered.) And write FU=—

Ax+A'x'+...,

BX+B'X

+ ..., ...

l

(3).

a a'

sxX "T R CD ~r . .„.•,

a!

,

btC -\~ b SC ~\~ • • • j

(4).

B'

The symbols iT, F, 7 possess properties which it is the object of this section to investigate.

12]

65

ON THE THEORY OF DETERMINANTS.

Let A, B,...,

A', B'',...,... A=

be given by the equations: ff,

-/,...

7,

a',

7",

8",

7".

8",

.(5).

It!

7 . (The upper or lower signs according as n is odd or even.) These quantities satisfy the double series of equations, Aa + 5 / 3 + . . . = « ,

.(6).

A'a+B'/3 + ... = 0,

&c. Aa + A'OL + . . . = « , ,

•(7),

B0 &c. the second side of each equation being 0, except for the r th equation of the r th set of equations in the systems. Let \ , fj,,... represent the rth, r + 1 th , ... terms of the series a, /3,... ; L, M,... the corresponding terms of the series A, B..., where r is any number less than n, and consider the determinant

A

,...L

•(8),

which may be expressed as a determinant of the nth order, in the form A

,...L

,0,

0

, o , o

,

0

c.

0,..

, o, o, 1,

, o,

1,

.(9).

66

[12

ON THE THEORY OF DETERMINANTS.

Multiplying this by the two sides of the equation (10),

a, a',

and reducing the result by the equation (©), and the equations (6), the second side becomes K,

0,...

0,

K,

(11),

o, o,

>,

i>(r+1),

which is equivalent to •(12), (r+l)

or we have the equation A

,... L

(13), H.(r+1)

I/(f

which in the particular case of r = n, becomes A, A',

B,... B',

•(14),

which latter equation is given by M. Cauchy in the memoirs already quoted; the proof in the " Exercises," being nearly the same with the above one of the more general equation (13). The equation (13) itself has been demonstrated by Jacobi somewhat less directly. Consider now the function FU, given by the equation (3). This may be expanded in the form A'x'+ ...)+ B (RS+BV+...) + ...]+ (R'f + s'v + ...) [A' (AX + A V + ...) + B' (RC + B V + .,.)+...] +

(15),

which may be written .(16),

12]

67

ON THE THEORY OP DETERMINANTS.

(in

by putting B = A (SA+S'A'+...)+B(SB

+

s'B'+...)

B' = A' (BA + a'A' + ...) + B ' (aB + s'B' + ...) + ..., .(18),

KFU= A, B , . A', B',

Hence,

A,

B,...

R,

S,...

A',

B',

B',

8',

A, A',

B, B',

(19),

or observing the equation (14), and writing A,

B,...

A',

B',

R,

S,

B',

S',

=J

.(20),

=r

•(21),

this becomes

KFU=Jf.(KU)n~1

.(22).

Hence likewise .(23).

Consider next the equation BiC + E ¥ + ...,

, A , A'

, ,

SX + S V +

B B'

1,

1,

(25).

A,

B,...

R,

S,

A',

B',

B\

S',

x,

£> A,

— rr — — ox

. (24).

x, A, B, x', A', B', v» B,

(26).

x', A', B', 9—2

68

[12

ON THE THEORY OP DETERMINANTS.

If the two sides of this equation are multiplied by the two sides of the equation (2), written under the form (27),

1, a,

/8,...

the second side is reduced to (28), X,

K

X',

.

=-J/./c^. U

• (29),

and hence t fJT T f*, ixv I V UTT\n—2 ± JJTPTT U ^ «/J. ) • UTT

(30).

Similarly FtU = Jr. (KU)n-^. U .(31); also combining these with the equations (22), (23),

1

J

KFU

l

K1U

. (32).

KU

It may be remarked here that if U, V are functions connected by the equation FV = cFV,

or 1U=c'iV,

(33),

I

then in general

If = cn~1V

(34).

To prove this, observing that the first of the equations (33) may be written I

FU=F(cn-1V) we have

V) * U=Jf[K

or

(35),

(c"-1 V)]n~2 c »-! V.

• (36),

• (37).

If neither / , / nor (KU) vanish, this equation is of the form

U=hV.

. (38),

whence substituting in (33), 1

=c

(39),

12]

69

ON THE THEORY OP DETERMINANTS.

which demonstrates the equation (34); and this equation might be proved in like manner from the second of the equations (33). If however, J = 0, or f = 0, the above proof fails, and if KU = 0, the proof also fails, unless at the same line « = 2. In all these cases probably, certainly in the case of KV = 0, n =}= 2, the equation (34) is not a necessary consequence of (33). In fact FU, or 'iJJ may be given, and yet U remain indeterminate. Let V'„ a,, /S/; . . . J . / ; B,, &c.... be analogous to U, a, / 3 . . . , At, B, &c.... and consider the equation

K{KUt.FU+gKU.FU) = K,A

t,

K/B+gicB,,

(40),

...

Multiply the two sides by the two sides of the equation (2), the second side becomes, after reduction, A.ji + Bfi + . . . ) ,

gic (A fa + Bf/3 +

gK (Afit + Bfi> + ...),

...),..

w + gK (A fa' + B/0' + . . . ) ,

Multiplying by the two sides of the analogous equation (42),

and reducing, the second side becomes KK,

(a, + go.),

3, + gfi),

.(43),

(44), whence

K(KU,.FU+gKU. FU) =

(45),

and similarly (46). In a similar manner is the following equation to be demonstrated, (47),

., "n...,

ap + a/a/..., Pp +/3>'....... a +ga ..., Pt +gP ..., a' +ga! ..., Pf +gp; ....

70

[12

ON THE THEORY OP DETERMINANTS. Suppose

£T=S (/>£ + o-*7 + . . . ) (ax + a'x' + ...)

(48),

this expression being the abbreviation of , (49),

[(n — 1) lines, or a smaller number]. KU=

then

2 a p,

2 a cr,.

2a'p,

2aV,

is = 0

(50),

which follows from the equation (©). Conversely, whenever KU=0, Also

U is of the above form.

FU = -

Aw+ A V + ...,

BX + B V + . . . , .

.(51),

Sa

R?+Si?+..., Sap

which may be transformed into Ax + A V ..., P

>

BX +

<*

B V ...,

(52),

,

(for shortness, I omit the demonstration of this equation). And similarly, 1U =

'os'+ ...,

SX+ SV

A%+Br)+ ..., a

A'£ + B'rj + ..., a'

where it is obvious that if the sum S contain fewer than (n — 1) terms, The equations (52), (53) express the theorem, that whenever KU=0, FU, 1U are each of them the product of two determinants. If next

ut=u+u,

•(53),

FU=0, the functions

12]

71

ON THE THEORY OF DETERMINANTS.

then in (45) taking g = — 1 [the Numbers (56) &c. which follow are as in the original memoir] K {K (U+U). FU- KU. F(U + U)} = K {K(U + U). VU- KU.1 (U +U)}

(56),

= (KUf-1 .(K(U+ U))"-1. KU, or observing the equation (50), )} = 0

(57).

Hence F {{K (U +U) .117- KU .1 (U+U)} = 1 {K (U+TJ). FU- KU. F (U+U)) are each of them the product of two determinants. But this result admits of a further reduction: we have F{K{U+

(58)

-.., a'|+/8/7?+...)

«,

-OL

«/ -a'

, 0 , - 0 , 01 +01

substituting at = a + Spa, &c...., also observing that if the second line be multiplied by x, the third by *', ... and the sum subtracted from the first line, the value of

i i i i i « HI Hi ii in i AH t • i iiii i mm — U, which in the expansion of the determinant is multiplied by zero: this may be written in the form -//

(KU)n~2(K(U+U))n~

... (59), at;+ 0i]+...,

tpa

Sera tea'

which may be reduced to Jf. (KU)™ .(K(U+ U))n-' x a'x' + ..., P

•(6.0),

(3x + /3V + . . . , ..
.

If each of these determinants are multiplied by the quantity (iff/)""1, expressed under the two forms A, A',

B,... B',

A,

A',

B,

B',

(61),

72

[12

ON THE THEOBY OF DETEEMINANTS.

they would become respectively KU.

,

x

KU.

...(62),

Ap+B
F{K(U+ U).

?U-KU.1(U+

(K(U+U)y \ KU )

-KU.F(U+

(63).

x

Ap+Ba+...,

The second side of this may be written under the forms

Kju+uy

A.X + AV + ...

KU

+ ...

A(Ap+B
multiplied into (64).

R R(Aa + A'a'..)+

S(Ba + B'a'..) + ...,

R'(Aa + A'a'..) + S'(Ba +

B'a'..)+...,

And ~p • • .

KU

R(Ap+B
y

) j * ~r O 30 ~r" • • •

> '

S(Ap+B<7..)+S'(A'p + B'a..)+...,

multiplied into

... (65). A (Aa + A'a'..)+B(Ba + B'a'..) + ..., A'(Aa +A'a'..) + B'(Ba+B'a'..) + ..., And again, by the equations (52), (53), in the new forms

x [(Aa + A'a' . . . ) ( R S + K V ...) + (Ba + B'a' ...)(sx + s V ...)...]]

/K(U+U)y{ )

(66),

.X {[(Ap + Bo- ... x [(Aa +A'a'...)

( x x + A'X' ...) +(Ba + B'a' ...)(BX

+ B'X'

...)...]} ...(67).

12]

73

ON THE THEORY OF DETERMINANTS.

Comparing these latter forms with the two equivalent quantities forming the first side of (53), and observing (33), (34), it would appear at first sight that U)

K(U+

x [(Aa + A'a! ...) (K# +

KV.

+ B'a' ...)

+ s'x'

K(U+U).FU-KU.F(U+U)

which however are not true, except for n = 2, on account of the equation (57). In the case of n = 2, these equations become

K(U+ U). VU-KU. W(U+ U) x [(Aa + AV ...)(

(68),

K(U+ U) FU-KU. F(U+ U) = [(Ap + Ba ...) (H^ + Si; ...)+ (A'p + B'a- ...) ( ...)...] x [(Aa + A'af ...)(kx+A'x' ...) + (Ba + B'a? ...

. ..)...]

(69),

and it is remarkable that these equations ((68), (69)) are true whatever be the value of n, provided 2 contains a single term only. The demonstration of this theorem is somewhat tedious, but it may perhaps be as well to give it at full length. It is obvious that the equation (69) alone need be proved, (68) following immediately when this is done. I premise by noticing the following general property of determinants. a + %p a, fi +%
The function .(70),

(where 2/oa = p1a1 + p3a2... + psas)> contains no term whose dimension in the quantities a, a' ..., or in the other quantities p, a..., is higher than s. (Of course if the order of the determinant be less than s or equal to it. this number becomes the limit of the dimension of any term in a, a'... or p, a ..., and the theorem is useless.) This is easily proved by means of a well-known theorem, %pa,

%
=0

(71),

Sjo'a, to-'a, C.

10

74

[12

ON THE THEOEY OF DETERMINANTS.

whenever s is less than the number expressing the order of the determinant. Hence in the formula (70), if % contain a single term only, the first side of the equation is linear in p, a,... and also in a, a',..., i.e. it consists of a term independent of all these quantities, and a second term linear in the products pa, pa',...aa, aa',... This is therefore the form of K (U+ U). Consider the several equations +B/3 + K = KU=Act = A'a' + B'/3' + ... = &c. it is easy to deduce ic, = K{U+U)=KU+Apa+B<m + A'pa' + B'aa'

(72),

(73).

To find the values of A, B, &c. corresponding to U+ U, we must write (74), = &c.

where

M'=

y",

8",...

*'= + S ll

i", 8'", M" =

0

,

S in S im

3'",.

i",

II


6

(75),

,

in

, &c.

mi

>

e ,

the order of each of these determinants being n — 2, and the upper or lower signs being used according as n — 1 is odd or even, i.e. as n is even or odd. Hence At = A + M' aa' + N' TO! +

(76),

and therefore KtA-KAt=

A2pa +(AB

) aa + (AC

)ra

+...

•(77),

+ AA' pa' + {AB' - KM.' ) aa' +(AC - «N' )ra' +... + AA"pa" + (AB" - KM") aa" + (AC" - «N") ra" +

the additional quantities G, T having been introduced for greater clearness. Now the equations AB' -KM

= A'B,

AC -KN'

=A'C,

AB" -

= A"B,

AG" -

= A"C,

KM"

KN"

(78),

12]

ON THE THEORY OF DETERMINANTS.

75

written under the form AB' -A'B

= KM',

AC

AB" -A"B = KM",

-A'O=KN\

(79),

A G" -A"G = KN",

are particular cases of the equation (13), and are therefore identically true. substituting in (77), KtA-KAt=

A%pa +ABaa+AGra

... +

Hence,

(80),

+ A A'pa' + A' Baa' + A' Gra ...+ + A"Apa" + A"Baa" + A"Gra" ...+ = (PA + aB + ...)(Aa + A'a' +...). Forming in a similar manner, the combinations K,B — KB/} ... K(A' — icA', KtB— KB/, ..., multiplying by the products of the different quantities Ax+A'x'..., Bx + B'x'..., ...

Rg + Sr)..., R'l+8'r).......

and adding so as to form the function

K(U+U).FU

— KU .F(U+ TJ), we obtain the required formula, viz. that the value of this quantity is ...)(K'Z+s'V...)+...] x [(Aa + A'af ...)(ASC+A.W...)

+ (Ba

(81);

+B'a'...)

with this theorem, I conclude the. present section,—noticing only, as a problem worthy of investigation, the discovery of the forms of the second sides of the equations (68), (69), in the case of £ containing more than a single term. § 2. On the notation and properties of certain functions resolvable into a series of determinants. Let the letters

rlt

r2,...rk

(1),

represent a permutation of the numbers 1, 2, ... k

(2).

Then in the series (1), if one of the letters succeeds mediately or immediately a letter representing a higher number than its own, for each time that this happens there is said to be a " derangement" or " inversion." It is to be remarked that if any letter succeed s letters representing higher numbers, this is reckoned for the same number s of inversions. Suppose next the symbol

denotes the sign + or —, according as the. number of inversions in the series (1) is even or odd. 10—2

76

ON THE THEORY OF DETERMINANTS.

[12

This being premised, consider the symbol

{P denoting the sum of all the different terms of the form A

i,i,...4/^(rI] the letters

rlt

r2...rk;

Prk
sly s2...sk;

(5),

&c

(6),

denoting any permutations whatever, the same or different, of the series of numbers (2) [and the several combinations of pa-... being understood as denoting suffixes of the A's]. The number of terms represented by the symbol (5) is evidently (1.2 ...k)n

(7).

In some cases it will be necessary to leave a certain number of the vertical rows p, a- ... unpermuted. This will be represented by writing the mark (•(•) immediately above the rows in question. So that for instance t t

f { ; i. Pkk

the number of rows with the (f) being x, denotes the sum of the (1.2 ...&)"-*

(9)

terms, of the form ±r ± s • • • Aprx °-Sl • • • #1$1 • • • Aprk (7^ . . . 0k<j)k

(10).

Then it is obvious, that if all the rows have the mark (•}•) the notation (8) denotes a single product only, and if the mark (•(•) be placed over all but one of the rows the notation (8) belongs to a determinant. It is obvious also that we may write t t p^

... 6-3-L ... (n)') = £ + „ + « , . . .

(Ap1o-1...dUiVi...(n) \

pkak

... 0k4>k

J

[

pkVk

where "Z refers to the different permutations, «!,

u2,...uk;

vlt

v3,...vk;

&c

(12),

which can be formed out of the numbers (2). The equation (11) would still be true, if the mark (*[") were placed over any number of the columns p, a-... Suppose in this equation a single column only is left without the mark (•{") on the second side of the equation; the first side is then expressed as the sum of a number (1.2 ... k)n'\ or generally ( 1 . 2 ... h)"-*-1

(13),

12]

ON THE THEORY OF DETERMINANTS.

77

of determinants, according as we consider the symbol (4) or the more general one (8). And this may be done in n or (n — x) different ways respectively. It may be remarked, that the symbol (8) is the same in form as if a single column only had the mark (•)•) over it; the number n being at the same time reduced from n to (n — x +1): for the marked columns of symbols may be replaced by a single marked column of new symbols. Hence, without loss of generality, the theorems which follow may be stated with reference to a single marked column only. Suppose the letters Pi,

P2,---Pk',

c i ,
(14)

denote certain permutations of «i> «2,...ak;

ft,

/3 2 ,.../3*; &c

.,

(15),

in such a manner that Pi

=

s

W

P* = ag>> — Pi = agk>

°"i=/9fi!>

o-2 = ^ Ai! , . . . o-i = /3 Ajt

(16).

Then the two following theorems may be proved: t ++

=±g±h ... (4a 1 if n be even: but in the contrary case t

t

) =+±g... (

By means of these, and the equation (11), a fundamental property of the symbol (3) may be demonstrated. We have

I"

i

J

{P

which when n is even, reduces itself by (17) to

= (At/31...(n))$(±g±g.l) t = 1.2...A (Aa1@1...(ny

(20)

78

ON THE THEORY OF DETERMINANTS.

[12

But when n is odd, from the equation (18),

'Aalt ft ...(n))

= (Aa1fi1...(n))"2l(±gl)

=0

(21),

since the number of negative and positive values of ±g are equal. From the equation (20), it follows that when n is even, the value of a symbol of the form ,

(n))

\ J

(22)

is the same, over whichever of the columns a, / 3 . . . the mark (f) is placed. denote this indifference, the preceding quantity is better represented by

To

this last form being never employed when n is odd, in which case the same property does not hold. Hence also an ordinary determinant is represented by All (24)

kk

'

the latter form being obviously equally general with the former one. It is obvious from the equations (17), (18), that the expression (22) vanishes, in the case of n even whenever any two of the symbols a. are equivalent, or any two of the symbols /3, &c; but if n be odd, this property holds for the symbols /3, &c, but ,not for the marked ones a. In fact, the interchange of the two equal symbols, in each case, changes the sign of the expression (22), but they evidently leave it unaltered, i.e. the quantity in question must be zero. Consider now the symbol (25), which, for shortness, may be denoted by t {A.k.2p}

(26).

12]

ON THE THEORY OP DETERMINANTS.

79

I proceed to prove a theorem, which may be expressed as follows: {A.k.2p}.{B.k.2q} = {AB\k.2p+2q-2} where

(27),

AB\rs,,iXy_ = 8. Ars,,,z Bxy,,x

(28),

the number of the symbols r, s,... being obviously 2p — l, and that of x, y,... being 2q — 1. The summatory sign 8 refers to I, and denotes the sum of the several terms corresponding to values of I from 1 = 1 to I = k. Also the theorem would be equally true if I had been placed in any position whatever in the series r, s... I; and again, in any position whatever in the series x, y... I, instead of at the end of each of these. With a very slight modification this may be made to suit the case of an odd number instead of one of the two even numbers 2p, 2q; (in fact, it is only necessary to place the mark (f) in [AB\ ...} over the column corresponding to the marked column in {A ...}, [A ...} being the symbol for which the number of columns is odd), but it is inapplicable where the two numbers are odd. Consider the second side of (27); this may be expanded in the form Sk...X0k...

(29),

where 2 refers to the different quantities s, ..., x, y,... as in (11). Substituting from (28), this becomes Z.Sll...Slic(+±s...±x±yA1Sl...k...AkSii...lk...BXll>i..jt...BXkVk^k)...(20). Effecting the summation with respect to x, y... this becomes

t.8h...8h

+ ±s.. A1Si...h...AkSk...h}

: [

(31), kk...lk)

2 now referring to s,... only. The quantity under the sign 2 vanishes if any two of the quantities I are equal, and in the contrary case, we have =±l{B.k.2q}

(32),

kk...lk) which reduces the above to t

{B.k.2q}.t

+ ±s...±lAl.Sl...k...Ak.Sk...lk

(33),

2 referring to the quantities s..., and also to the quantities I. And this is evidently equivalent to t t {A.k.2p}{B.k.2q} (34), the theorem to be proved. It is obvious that when p=l, q=l, the equation (27), coincides with the theorem (©), quoted in the introduction to this paper.

80

[13

13. ON THE THEORY OF LINEAR TRANSFORMATIONS. [From the Cambridge Mathematical Journal, vol. IV. (1845), pp. 193—209.] THE following investigations were suggested to me by a very elegant paper on the same subject, published in the Journal by Mr Boole. The following remarkable theorem is there arrived at. If a rational homogeneous function U, of the nth order, with the TO variables x, y..., be transformed by linear substitutions into a function V of the new variables, £, i)... \ if, moreover, 6U expresses the function of the coefficients of U, which, equated to zero, is the result of the elimination of the variables from the series of equations dxU=0, dyU=0, &c, and of course 6V the analogous function of the coefficients of V: then 6V = Ena.611, where E is the determinant formed by the coefficients of the equations which connect x, y... with £, r)..., 'and a = (ra — 1)™r-1. In attempting to demonstrate this very beautiful property, it occurred to me that it might be generalised by considering for the function U, not a homogeneous function of the nth order between m variables, but one of the same order, containing n sets of TO variables, and the variables of each set entering linearly. The form which Mr Boole's theorem thus assumes is 6 V = Ef. E2a ... Ena. 8 U. This it was easy to demonstrate would be true, if 6U satisfied a certain system of partial differential equations. I imagined at first that these would determine the function 6U, (supposed, in analogy with Mr Boole's function, to represent the result of the elimination of the variables from dXlU=0, dVlU = 0,... dx%U — 0, &c.): this I afterwards found was not the case; and thus I was led to a class of functions, including as a particular case the function 8TJ, all of them possessed of the same characteristic property. The system of partial differential equations was without difficulty replaced by a more fundamental system of equations, upon which, assumed as definitions, the theory appears to me naturally to depend; and it is this view of it which I intend partially to develope in the present paper. 1 The value of a was left undetermined, but Mr Boole has since informed me, he was acquainted with it at the time his paper was written; and has given it in a subsequent paper.

13]

81

ON THE THEORY OF LINEAR TRANSFORMATIONS.

I have already employed the notation a. ,

/3 ,

7 ,

«', P, i,

8 ,.

(1)

8',

a", P', j", 8", (where the number of horizontal rows is less than that of vertical ones) to denote the series of determinants, a,

(2).

/3, 7 .

a', P,

7,

which can be formed out of the above quantities by selecting any system of vertical rows; these different determinants not being connected together by the sign +, or in any other manner, but being looked upon as perfectly separate. The fundamental theorem for the multiplication of determinants gives, applied to these, the formula A , B ,

A', A", where

C , D , ...

=E

B', C, jy, B", G", B",

« , /8 , 7 . } , «', P,

7,

•(3),

8'.

a", P', 7". 8",

A =\<x +\'a' +\"a"

(4),

&c.

•(5).

v, / , and the meaning of the equation is, that the terms on the first side are equal, each to each, to the terms on the second side. This preliminary theorem being explained, consider a set of arbitrary coefficients, represented by the general formula ™t

(6),

in which the number of symbolical letters r, s,... is n, and where each of these is supposed to assume all integer values, from 1 to m inclusively. C. 11

82

[13

ON THE THEORY OF LINEAR TRANSFORMATIONS.

Let

««/*/... , as/%"...,

•(7)

represent the whole series, taken in any order, in which the first symbolical letter is a. Similarly, rt;at „ ' . . . ,

(8),

the whole series of those in which the second symbolical letter is a, and so on. Imagine a function u of the coefficients, which is simultaneously of the forms u=Hn

la/*/..., 2S;t;...,

ls/'t/'-. 2«/'«/'•••,

(A),

r,;%%,', ..., r,,"2 *„"•••> &c.; in which Hp denotes a rational homogeneous function of the order p. The function H is not necessarily supposed to be the same in the above equations, and in point of fact it will not in general be so. The number of equations is of course —n. The function u, whose properties we proceed to investigate, may conveniently be named a " Hyperdeterminant." Any function satisfying any of the equations (A), without satisfying all of them, will be an " Incomplete Hyperdeterminant." But, considering in the first place such as are complete— Let rst... be a new set of coefficients connected with the former ones by a system of equations of the form rst... =\1rlst...+

Xf2,st...+\mrmst

(9),

(where the r in \ / . . . is not an exponent, but an affix). Suppose u is the same function of these new coefficients that u was of the former ones. Then consider the first of the equations (A) and the equation (3), and writing (10),

L= V, V ,

we have immediately the equation (11). Consider the new set of coefficients rst ... = fiisrlt...

+ fi2sr2t ... + ...

rrnt,

(12),

13]

83

ON THE THEORY OF LINEAR TRANSFORMATIONS.

and u the analogous function of these; then, from the second of the equations (A) and the equation (3), and writing (13),

(14).

.u In like manner, considering the new coefficients rsi ... , where rsi ... = Vi rsi ... + v} rs2

+ vn( r'sm ...

(15),

the new function u, and the quantity N given by (16),

we have, as before, (17), or

(18) ;

u=

whence, generally, denoting the last result by u', (B, 1). Consider now the function SS2... (rst...wryszt..:)

(19),

where the %'s refer successively to r, s, t, ..., and denote summations from 1 to m inclusively. If u be looked upon as a derivative from the above function, we may write ... (rst... xryszt...) Assume

xr =

x2...

+

(20).

Xrm®m

(21). It is easy to obtain ... (rst.,..ocryszt...)

— %%% ... (rst ...xryszt...),...

= S 2 S ... (rsi ... xryszt...)

(22),

and the formula for (u) becomes © 2 2 2 . . . (rsi...xryszt)...

= LPMPNP . . . © 2 2 2 ...(rst...

xryszt)

Proceeding to obtain the expression of the coefficients rsi... coefficients rst..., we have ...)

(B, 2).

in terms of the (C),

84

ON THE THEORY OF LINEAR

[13

TRANSFORMATIONS.

where the 2's refer successively to f, g, h,... , denoting summations from 1 to m inclusively. Having this equation, it is perhaps as well to retain u' =D>MPNP ...u

(B, 1),

instead of (B, 2), that form being principally useful in showing the relation of the function u to the theory of the transformation of functions. It may immediately be seen, that in the equations (B), (C) we may, if we please, omit any number of the marks of variation (•), omitting at the same time the corresponding signs 2, and the corresponding factors of the series L, M, If ... Also, if u be such as only to satisfy some of the equations (A); then, if in the same formulte we omit the corresponding marks ('), summatory signs, and terms of the series L, M, N... , the resulting equations are still true. From the formulae (A) we may obtain the partial differential equations 2 2 ... last... -r-: V 2 2 ...(rat \

I u = 0, or pu,

(D),

dpst...) ...

d ™ - } u = 0, or pu, r drfit..../ 1

according as a. is not equal, or is equal, to /3; and so on : the summatory signs referring in every case to those of the series r, s, t,..., which are left variable, and extending from 1 toTOinclusively. To demonstrate this, consider the general form of u, as given by the first of the equations (A). This is evidently composed of a series of terms, each of the form cPQR...(p Is/*/...,

in which

factors),

l s / ' i , " . . . . . . . (m terms)

Q, R, &c. being of the same form; and we have d .,-=—— , u = cQR... 2 2 ...last... ast... dfist...) \ and

22...

d

''' dfist...

Is/*/ ...,

d J 5 - T — P + &C.+ &C, dfist...

I s / ' * / ' . . . . . . . (m terms)

<*/..., «/'*/'

= 0:

13]

ON THE THEOKY OP LINEAR TRANSFORMATIONS.

85

so that all the terms on the second side of the equation vanish. If, however, /3 = a, 22...

\

dfist ...)

whence, on the second side, we have cQR...P

+ cPR ...Q + &Dc.=p.cPQR...+&c.

+&c.

or the theorem in question is proved. In the case of an incomplete hyperdeterminant, the corresponding systems of equations are of course to be omitted. In every case it is from these equations that the form of the function u is to be investigated; they entirely replace the system (A). A very important case of the general theory is, when we suppose the coefficients rst... to have the property r's't' ...= r"s"t" ..., whenever r's'tf ... and r"s"t" ... denote the same combination of letters; and also that the coefficients X are equal to the coefficients ji, v ... , each to each. In this case the coefficients rst... have likewise the same property, viz. that r"s"i" ... = r's't'..., whenever r's't' ... and r"s"t" ... denote the same combination of letters. The equations (B, 1), (B, 2), become in this case u' = LnP.u

(B, 3),

where only different combinations of values are to be taken for r, s, t, ... and a, yS,... express how often the same number occurs in the series. In the equation (C), fi, v must be replaced by X, the equations (D) are no longer satisfied; the equations (A) reduce themselves to a single one, (so that there can be no question here of incomplete hyperdeterminants) : but this is no longer sufficient to determine the function sought after. For this reason, the particular case, treated separately, would be far more difficult than the general one; but the formulae of the general case being first established, these apply immediately to the particular one1. The case in question may be defined as that of symmetrical hyperdeterminants, (a denomination already adopted for ordinary determinants). It would be easily seen what on the same principle would be meant by partially symmetrical hyperdeterminants. I have not yet succeeded in obtaining the general expression of a hyperdeterminant: the only cases in which I can do so are the following: I. p = 1, n even, (if n be odd, there only exist incomplete hyperdeterminants). II. p = 2, m = 2, n even. III. p = 3. m = 2, « = 4. I. The first case is, in fact, that of the functions considered at the termination of a paper in the Cambridge Philosophical Transactions, vol. vnr. part i. [12]; though at that time I was quite unacquainted with the general theory. 1

See concluding paragraph of this paper.

86

ON THE THEORY OP LINEAR TRANSFORMATIONS.

[13

Using the notation there employed, we have t u= ( 1 1 . . . 0 ) 22 mm ) a complete hyperdeterminant when n is even; and when n is odd the functions t

11...(») ) , 22

22

mm ) ( mm are each of them incomplete hyperdeterminants. (A) In the case of n = 2, the complete hyperdeterminant is simply the ordinary determinant 11, 12,... lm 2 1, 2 2, ...2m ml, m2, ...mm j Stating the general conclusion as applied to this case, which is a very well known one, "If the function £7 = 2 2 (rs. xrys) be transformed into a similar function by means of the substitutions xr =

2

^yx + then

ii, i2,... = V, 21, 22, V,

1

V ^ + x, a ...+X ^y*...+t* smym; fi2\...

ii,

12,...

21,

22,

Also, by what has preceded, fi = 2 2 (A/./V ./ysa), r, s extending as before, from 1 to m inclusively, the determinant itself is the product of three determinants ; the first formed with the coefficients rs, the second with the quantities x, and the third with the quantities y."

13]

ON THE THEORY OF LINEAR TRANSFORMATIONS.

87

In a following number of the Journal I shall prove, and apply to the theories of Maxima and Minima and of Spherical Coordinates, (I may just mention having obtained, in an elegant form, the formuliB for transforming from one oblique set of coordinates to another oblique one) the more general theorem, " If k be the order of the determinant formed as above, the determinant itself is a quadratic function, its coefficients being determinants formed with the coefficients rs, its variables being determinants formed respectively with the variables x and the variables y; and the number of variables in each set being the number of combinations of k things out of m, ( = 1 if k = m; if Jc>m the determinant vanishes)." I shall give in the same paper the demonstration of a very beautiful theorem, rather relating, however, to determinants than to quadratic functions, proved by Hesse in a Memoir in Grelles Journal, vol. xx., '' De curvis et superficiebus secundi ordinis;" and from which he has deduced the most interesting geometrical results. Another Memoir, by the same author, Crelle, vol. xxviil., " Ueber die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiter Grade mit zwei Variabeln," though relating in point of fact rather to functions of the third order, contains some most important results. A few theorems on quadratic functions, belonging, however, to a different part of the subject, will be found in my paper already quoted in the Cambridge Philosophical Transactions [12]; and likewise in a paper in the Journal, Chapters in the Algebraical Geometry of n dimensions [11]. I shall, just before concluding this case, write down the particular formula corresponding to three variables, and for the symmetrical case. It is, as is well known, the theorem, U = Axs + By* + Gz2 + 2Fyz + 2Gxz + 2Hxy

"If be transformed into by means of

then

($13® - &J p - 2 W - 0£?$' + (a/3'7" - a/3'V + a'/3"7 - <x'/3y" + a"/37' - a"/3'7)2 (ABC - AF* - BG* - CH* + 2FGH)." (B) Let n = 3, and for greater simplicity m = 2 ; write

so that

6=211, c = 121,

/=212, # = 122,

d = 221,

h = 222,

U = a %$& + b x^y^ + c x^y^ + d x$& + e a^y^ +fx2y1z!s + g x^y^ + h x^y2z2.

88

ON THE THEORY OF LINEAR TRANSFORMATIONS.

There is no complete hyperdeterminant (i.e. for p = 1), and the incomplete ones are ah — bg — cf + de = ut , suppose, ah — de — bg + cf=

ult,

ah — cf — de + bg = ut//.

Thus, suppose the transforming equations are

x1 = V xx + V x2,

x + Mi

Z\ = v? ij + i/j3 i 2 , ^2 = v£ z\ + j/ 2 2 i 2 ; 'u/ — (fii1 fii — ^fh2)

then

(vi1 v£ ~ vi 2

v

i)

u

i > where y, z are changed

1

w „ = W vi - v? vf) ( V X2 - X2 V ) utl ,

«'„, = (^ l x* ~ V V ) W M22- K tf) um,

„

„

We might also have assumed u, —ad— be, or eh — gf, un = af — be, or ch — dg, %

m = ag — ce, or bh — df,

but these are ordinary determinants. {€)

« = 4, m = 2.

a =1111,

i = 1112,

6 = 2111,

j = 2112, jfc=1212,

e=1121, /=2121, g = 1Zll, h = 2221, U—

m=1122, rc=2122, o=2212, ^9=2222.

a x^y^w^ + b x^y^z1w1 + c xlyiz1w1 + +fx11y1zssw1 + g x2ysz1 y x zx + m xiy1ziw2 + n xjy&Wt + 0

we have M

=

ap — bo —on + dm — el +fk + gj — hi;

z, x

x, y

[13

13]

ON THE THEORY OF LINEAR TRANSFORMATIONS.

89

so that, with the same sets of transforming equations as above, and the additional one,

'u = (A^Xa2 — X21X12) (/h1/*/ — ^^)

we have

(vi-vf — v^vf) ipipi — pipi)•

M

j

this is important when viewed in reference to a result which will presently be obtained. If we take the symmetrical case, we have U =ax4 +4/3 a?y + 67 afy" + 4<S%ys + e y^; which is transformed into U' = aV4 + 4/3Vy +fy'cc'Y*+ kh'x'y'3 + e'y'\ by means of = Xx' + /J, y', u = a e - 4/3 8 + 3 7 2 ,

then, if

fiy.u.

we have II.

Where p = 2, m = 2, n is odd.

The expression

u=

t

t f l l l l ... [1222 t f l l l l ... 2222

[2222 t J2111 .. 12222 is a complete hyperdeterminant; and that over whichever row the mark (f) of nonpermutation is placed. The different expressions so obtained are not, however, all of them independent functions: thus, in the following example, where n = 3, the three functions are absolutely identical. (A).

n = 3, notation as in I. (B).

u = a?h? + by + c2/2 + d?e2 - 2ahbg - 2ahcf- 2ahde - 2bgcf- 2bgde - 2cfde + badfg + 4bech, and then

u = (VV - \JV)2 (frW ~ f^WT ("iV - "iWY «•

This is in many respects an interesting example. may be expressed in the three following forms:

We see that the function v

u = (ah-bg - cf+ de)2 + 4 (ad -be) (fg - eh)

(1),

2

(2),

u = (ah-bg-de

+ c/) + 4 (af - be) (dg - ch) (ag - ce) (df-bh)

C.

(3), 12

90

ON THE THEORY OF LINEAR TRANSFORMATIONS.

[13

which are indeed the direct results of the general form above given, the sign (•)•) being placed in succession over the different columns: and the three forms, as just remarked, are in this case identical. We see from the first of these that u is of the second or third, from the second that u is of the first or third, from the third that u is of the first or second of the three following forms: u= H2

a, b, c, d

a, b, e, f

e, f, g, h

c, d, g, h

a, c, e, g b, d, f, h

which is as it should be. The following is a singular property of u. .bet

,

du

,,

a=*T-,

,,

.du

O = + -JT,

, du ^dh'

k=

then, u' being the same function of these new coefficients that u is of the former ones, u' = u3. To prove this, write p = ah — bg — cf+ de, q = (ad — be), r = eh —fg; at = ap — 2qe, b, = bp - 2qf,

et = — 2ra + pe, /, = - 2rb +pf,

c,=cp-2qg,

g, =

-2rc+pg,

dl = dp-2qh,

hl =

-2rd+ph;

we have, as a particular case of the general formula just obtained, «, = (p? — iqr)2 u = u*. M, = M3.

Also

a, =

h',

e, =

d',

d t=

e',

A, =

a';

s

whence ut = u', that is v! = u . There is no difficulty in showing also, that if a", b", ... h" are derived a', b'... h', as these are from a, b, ... A, then

from

a" = u2a, b" = u% ...h" = u'h. The particular case of this theorem, which corresponds to symmetrical values of the coefficients, is given by M. Eisenstein, Orelle, vol. xxvu. [1844], as a corollary to his researches on the cubic forms of numbers. Considering this symmetrical case U= aa? + u = a?& -

13]

ON THE THEORY OF LINEAR TRANSFORMATIONS.

91

so that if U be transformed into U' = aV s + 3/3VY + 3 7Vt/'2 + By, by means of oc=\oo'+fiy', 2/= *•/+/*/> «' = a'23'2 - 6a'g'/3'7' - 3/8V + 4/3'3S' + 4«V> then if w' = (X/A, — X^)6. u. we have III. Notation as in I. (G),

p = 3 , m = 2, w = 4.

where A, B are arbitrary constants, and gj, 23, &c.... p j are functions of the coefficients, given as follows :— gl = asp3 - bsos - c3n3 + d3m3 - eH3 +/ 3 & 3 + g3j3 - hH3. J3 = -

a?p*bo + b2o2ap + cVdm a?fen + fro2dm + cVap aPpH + VoPfk + cVgj tffM + &o2gj + cV/k

- dWcn - d2m?bo - d2m>hi - d2m?el

+ e2l2fk - f2k2el + ePPgj - fWhi + &Pap - pWbo + eH2dm - f2k2cn

- g^fhi + hH?gj - g2fel + hH'fk - tffen + hH*dm - g2fbo + hH*ap.

<& = + tffdm - tVcra - cVbo + dam*ap - eH2M + fk*gj + g2ffk - hH2el - eH%o + fWap + g*fdm - hH2cn -WcPel - tiWhi + dVgj - bWhi - cVel + d*m*fk - e2l2cn + / ! W B + gaj2ap - hH%o. 3D = apbocn — apbodm — apendm +bocndm — elf kgj + elfkhi + elgjhi — higjfk + apboel — apbo/k — cndmgj + dmcnhi — elfkap + elfkbo + gjhicn — higjdm — apbogj + apbohi + endmel — dmcnkf+ elf ken — elfkdrn — gjhiap + higjbo + apcnel — bodmfk — cnapgj + dmbohi — elgjap +fkhibo + gjelcn — hifkdm — aponfk + bodmel + cnaphi — dmboel + elgjbo — fkhiap — gjeldrn 4- hifken — apdmel + boenkf +cmbogj — dmaphi + elhia/p — fkgjbo —gjfkcn + hidmel. (Q. = apdmjk — bocnel — bocnih + dmapgj — elhibo +fkapgj 2

2

+ gjfkdm — Melon.

jp = d>phjo — tygoip — &nflm + d?emkn — e dlkn + f ckml + g2bjpi — h?aijo — Pphbg + fogah + k2nfde — Pmecf + mHdcf — n2ckde — cPbjah + p*aibg + a?phkn — tygolm — c2nfip + d?emp — e?dljo + f2ckip + g2bjpl — h?aink — i2phcf + fogde + l&nfah — Pmebg + mHdbg — tfekah — o2bjde + p^aicf + a2phlm — tygokn — c2nfjo + d2emip — eHdpi + fackjo + g*bjnk — h2aiml — fiphde + fgocf + k^nfbg — l2meah + m2dlah —ri*ckbg— cPbjcf + p2aide + a2pdno — b2cmpo —
92

ON THE THEORY OF LINEAR TRANSFORMATIONS.

[13

(ffif = apbgkn — boalhm — cndiep + dmcjfo — elfocj +fkpied +gjhmla— ignbk — ihjocf + gjidep + kflamh — lekbng + mdnbkg — ncmhal — obpied + paofjc + apbglm — boahkn — cndejo + dmcfip — elcfip +fkedoj + gjhakn— hibgml — ihjode + gijpcf + kflmbg — leknah + mdknah — ncmlbg — obpicf + paqjde + apcflg — bojehk — cnjehk + dmilgf — elmpbc + fkadno + gjadno — hipmcb — ihadno + gjbmcp + kfcbpm — eladno + mdjehk — ncilfg — obilfg + pahejk + apcflm — bodkne — cnahjo + dmbgip — elbgip +fkahjo + gjednk — hicfml — ihknde + mdahjo + kfipbg — bocfml + gjcfml — ncpigb — leahjo + paknde + apidng — bojcmh — cnbkpe + dmaflo — elmhjc — fkidng + gjaflo — ihbkpe — ihqflo + gjbkpe + fkcjhm — elidng + mdbkpe — cnaflo — boidng + pahmcj + apidfo — bojcep — nbkhm + cdmalng — elmnbk —fkalng + gjidfo — ihpecj — ihalng + gjkbmh + fkcjpe — elidfo + mdjcep — ncidfo — obalng + pahmbk. yfy = a?honl — ¥pgmk — &pfmj + d?onie — tfdpkj + filco + g^blni — h?ainkj — tfpdfg + j2oech + Jt?nbeh - Pmafg + ntfblch - ri*kadg — o2jadf + pHbec: we have, as usual,

u = ( v v - v v ) 3 OhW - frW)s (»iW - "2V)3 wp? - P1W)3 • «• Particular forms of U are 4 = 1, 5 = 0: u = @i + 3'& + 3<&+G'& + 6<& = (ap-bo-cn + dm-el+fk + gj-hiy, =v suppose. A = l, B = 9: u = & + 333 - 6© - S® + 33© + 9 # - 18® - 27^, =*6U suppose, where 0U=0 is the result of the elimination of the variables from the equations dXlU=0, dyiU=O, dZlU=0,
where

'u =
M = ( W - VV) 3 (fhW ~ frWY OiV - "2V)3 (fhW ~

13]

ON THE THEORY OP LINEAR TRANSFORMATIONS.

93

v3 = Mvs,

and thence

which coincides with a previous formula, and

6 U=M0U: whence, eliminating M,

an equation which is remarkable as containing only the constants of U and U: it is an equation of condition which must exist among the constants of U in order that this function may be derivable by linear substitutions from TJ. In the symmetrical case, or where U = ascl + 4>fix?y 4- Qyafy* + 4S«y + ey*,

it has been already seen that v is given by i> = ae-4/3S+3 7 2 . Proceeding to form 8JJ, we have - a2e2/38 + 372/32S2 - 3/98-y4), = 3 (a 2 ey + 2 7 6 + aerf - 4/3383), = 6( JF = 6 (a*8V + e'/SV - 2/33e7S - 2S 3 a/3 7 - 4/3yS 2 . + 4 . ^SST4 + 73/32e + 7 3 ag 2 ), CS = 12 (ae/88 7 2 - a/3 7 S 3 - 678/33 + y'aS 2 + 7 3 e^ 2 + ySSv4 - 2/3272S2), ^ = (a2S4 + e2/34 - 4/32e73 - 4a 7 3 8 2 + 6/S272S2), and these values give 6U= ahs- §a&&e-

12a2y8Se2 - 18a272e2 - 27a2S4 - 27/84e2 + 36/32
so that this function, divided by (ae — 4/3S + 372)3, is invariable for all functions of the fourth order which can be deduced one from the other by linear substitutions. The function ae — 4/38 + S^f occurs in other investigations: I have met with it in a problem relating to a homogeneous function of two variables, of any order whatever, a, /3, y, 8, e signifying the fourth differential coefficients of the function. But this is only remotely connected with the present subject. Since writing the above, Mr Boole has pointed out to me that in the transformation of a function of the fourth order of the form aa? + 4
94

ON THE THEORY OF LINEAR TRANSFORMATIONS.

[13

of the third order ace — b2e — ad2 — c3 + 2bdc, possessing precisely the same characteristic property, and that, moreover, the function 6u may be reduced to the form (ae - 4
the latter part of which was verified by trial; the former he has demonstrated in a manner which, though very elegant, does not appear to be the most direct which the theorem admits of. In fact, it may be obtained by a method just hinted at by Mr Boole, in his earliest paper on the subject, Mathematical Journal, vol. u. p. 70. The equations dx2u = 0, dxdyu = 0, dyht = 0, imply the corresponding equations for the transformed function: from these equations we might obtain two relations between the coefficients, which, in the case of a function of the fourth order, are of the orders 3 and 4 respectively: these imply the corresponding relations between the coefficients of the transformed function. Let A=0, B=0, A'=0, B' = 0, represent these equations; then, since A = 0, 5 = 0, imply A' — O, we must have A' — KA' + MB, A, M, being functions of X, \', \x, &c. fi': but B being of the fourth order, while A, A' are only of the third order in the coefficients of u, it is evident that the term M.B must disappear, or that the equation is of the form A' = AA. The function A is obviously the function which, equated to zero, would be the result of the elimination of x2, xy, y2, considered as independent quantities from the equations ax2 + 2bxy + cy2 = 0, bx2 + 2cxy + dy2 = 0, ex2 + 2dxy + ey2 = 0, viz. the function given above. Hence the two functions on which the linear transformation of functions of the fourth order ultimately depend are the very simple ones ae — 4bd + 3ca,

ace - ad2 - eb2 - c3 + 2bdc,

the function of the sixth order being merely a derivative from these. The above method may easily be extended: thus for instance, in the transformation of functions of any even order, I am in possession of several of the transforming functions; that of the fourth order, for functions of the sixth order, I have actually expanded: but it does not appear to contain the complete theory. Again, in the particular case of homogeneous functions of two variables, the transforming functions may be expressed as symmetrical functions of the roots of the equation u = 0, which gives rise to an entirely distinct theory. This, however, I have not as yet developed sufficiently for publication. There does not appear to be anything very directly analogous to the subject of this note, in my general theory: if this be so, it proves the absolute necessity of a distinct investigation for the present case, that which I have denominated the symmetrical one.

14]

95

14 ON LINEAE TRANSFORMATIONS. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 104—122.] IN continuing my researches on the present subject, I have been led to a new manner of considering the question, which, at the same time that it is much more general, has the advantage of applying directly to the only case which one can possibly hope to develope with any degree of completeness, that of functions of two variables. In fact the question may be proposed, " To find all the derivatives of any number of functions, which have the property of preserving their form unaltered after any linear transformations of the variables." By Derivative I understand a function deduced in any manner whatever from the given functions, and I give the name of Hyperdeterminant Derivative, or simply of Hyperdeterminant, to those derivatives which have the property just enunciated. These derivatives may easily be expressed explicitly, by means of the known method of the separation of symbols. We thus obtain the most general expression of a hyperdeterminant. But there remains a question to be resolved, which appears to present very great difficulties, that of determining the independent derivatives, and the relation between these and the remaining ones. I have only succeeded in treating a very particular case of this question, which shows however in what way the general problem is to be attacked. Imagine p series each of m variables *i,

yx.-.&c.

aoa, y2,...&c.

xp,

yp,...&c,

where p is at least as great as m. Similarly p* series each of nC variables a^,

ya\...&c.

Xt,

y * , . . . , &c.

...xp-,

y \ ; . . . &c,

96

[14

ON LINEAR TRANSFORMATIONS.

pv at least as great as m\ and so on. Let the analogous variables x, y. nected with these by the equations x =\x

be con-

+ fi y + ...,

y =X'i x' = Xv x + /A y' + ...,

w h e r e x , y , . . . s t a n d f o r X i , y l t . . . o r # 2 > 2/a. ••• ° r xp> y P , • • • ' , & , y \ - - - s t a n d K

y

or Xz, 2/ 2 \... or x p, y p\ &c The coefficients X, //.,..., X', p!,... remain the same in all these systems. Suppose next,

f o rx

&c.; X\ p,...,

x

\ yi,...

X",

/A'\...

i.e.

(where S^, Sj,... are the symbols of differentiation relative to x, y, &c). Then evidently

with similar equations for $\ i]\...

Suppose

that is to say ||ilj| is the series of determinants formed by choosing any m vertical columns to compose a determinant, and similarly ||1T||, &c. Suppose, besides,

E=

X,

fj,,...

X,

fj,,...

Then, by the known properties of determinants,

PI|=^||O||, ||^||=^||^|| &c, i.e. the terms on the one side are respectively equal to the terms on the other. Hence if

i.e. • a rational and integral function, homogeneous of the order / in the quantities of the series ||U||, homogeneous of the order / v in the quantities of the series ||O%||, &c, we have immediately

14]

ON LINEAR TRANSFORMATIONS.

or if U be any function whatever of the variables x, y... the linear substitutions above into JJ, then

or the function

97 which is transformed by

• U

is by the above definition a hyperdeterminant derivative. The symbol • may be called " symbol of hyperdeterminant derivation," or simply " hyperdeterminant symbol." Let A, B,... represent the different quantities of the series [|12||,—Ay, By,... those of the series ||fT||, &c. ..., then • may be reduced to a single term, and we may write

Also U may be supposed of the form

where ©, <S> are functions of the variables of one of the sets x, y,..., sets xx, 2/\ ..., &c, thus ® is of the form F(xu

ylt...xx\

of one of the

yi,...),

and so on. The functions ©, <E>... may be the same or different. It may be supposed after the differentiations that several of the sets x, y,...' or of the sets x\ yy, ... become identical: in such cases it will always be assumed that the functions ©,... into which these sets of variables enter, are similar; so that they become absolutely identical, when the variables they contain are made so. Thus the general expression of a hyperdeterminant is

• U= A«Bt... A in which, after the differentiations, any number of the sets of variables are made equal. For instance, if all the sets x, y ... and all the sets x\ y ... are made equal, the hyperdeterminant refers to a single function F(x, y...x\ yx...). In any other case it refers not to a single function but to several. What precedes, is the general theory: it might perhaps have been made clearer by confining it to a particular case: and by doing this from the beginning, it will be seen that it presents no real difficulties. Passing at present to some developments, to do this, I neglect entirely the sets x\ y\.. and I assume that the number m of variables in each of the sets x, y ... reduces itself to two; so that I consider functions of two variables x, y only. The functions ©, <3>, &c. reduce themselves to functions Vlt V2...VP of the variables xx, yu or x2, y2... or xp, yp. Writing also £i»72- & ' ? i = 1 2 , &c.

the symbols A, B... reduce themselves to 12, 13 Hence for functions of two variables, there results the following still tolerably general form • £f = 12a 13P T4Y...2T3^' 24 y '...84 y "... V.V.V.V,... C.

13

98

ON LINEAR TRANSFORMATIONS.

[14

The functions F,, V^... may be the same or different: but they will be supposed the same whenever the corresponding variables are made equal. This equality will be denoted by writing, for instance,

nvwvv... to represent the value assumed by

nv1v2vsvi... when after the differentiations •*"i, 2/i = #s, 2/3 = #4, 2/4 = «, y ;

x2,

y 2 = x', y';

&c.

It is easy to determine the general term of • £/.

To do this, writing for shortness

a H-/3+7 . . . = / , ,

&c. N=(_Y+s+t...+,+*...r...W

iM W

or Kf^yrV=

[£7 [7?'

'' - [t'j

V'r or V-\

the general term is y <»•+»+«••• •p'.a-r+s'+f... y ,/3-«+(3'-s'+r.

where r, s, i, ...s', t',...t",... extend from 0 to a, /3, 7.../S', 7', . . . 7 " . . . respectively. It would be easy to change this general term in a way similar to that which will be employed presently for the particular case of n F , F 2 F 3 . If several of the functions become identical, and for these some of the letters / are equivalent, it is clear that the derivative QZ7 refers to a certain number of functions Vlt F 2 . . . the same or different, of the variables oc, y; x', y'';... and besides that this derivative is homogeneous, of the degrees 6lt #/,... with respect to the differential coefficients of the orders flt f x ' , . . . &c. of Vlt (consequently homogeneous of the order 01 + O1' +... with respect to these differential coefficients collectively), homogeneous and of the degrees 62, 02',... with respect to the differential coefficients of the orders / 2 , / / . . . of F 2 , (consequently of the order 0 2 + # / . . . with respect to these collectively), and so on. The degree with respect to all the functions is of course 61 + 61 ... + 02+02 + •••,—p suppose. In general, only a single function will be considered,

14]

ON LINEAR TRANSFORMATIONS.

99

and it will be assumed that • U only contains the differential coefficients of the fth order. In this case, the derivative is said to be of the degree p and of the order f. The most convenient classification is by degrees, rather than by orders. Commencing with the simplest case, that of functions of the second order (and writing V, W instead of V1, V?), we have O F F =12° VW, (where £, % apply to V and f2, r)3 to W). the sequel by the notation

This will be constantly represented in

T2aVW = Ba(V, W). Hence, writing we have

8X«V=V-\ V" 1 ^V = V-1 ... , Ba(V,

W)= V'»W'«- ^

V-1W-'^1+ ...;

and in particular, according as a is odd or even, Ba(V,

J5(F continued to the term which being divided by two.

V) = 0,

F ) F « F « F contains

1

F »

1

+

V'^V'i"-, the coefficient

of this last

term

Thus, for the functions \ {aa? + 2bxy + ctf), •^{aaf + 4>ba?y + 6cafy2 + 4
which have all of them the property of remaining unaltered, & un facteur pres, when the variables are transformed by means of x = \x + /My, y = ~k'x + fi'y. Thus, for instance, if these equations give a«2 + tbxy + cy2 = dx2 + 2bd;y + cy\ then

ac - 62 = ( V - X » 2 . (ac - b%

and so on. This is the general property, which we call to mind for the case of these constant derivatives. 13—2

100

ON LINEAR TRANSFORMATIONS.

[14

The above functions may be transformed by means of the identical equation W) = Ua'kBk(V, W),

Ba(V,

to make use of which, it is only necessary to remark the general formula

Thus, if k = 1, we obtain for the above series, the new forms ac — b2, (ae-bd)-S(bd-c2), (ag - &/) - 5 (bf - ce) + 10 (ce - d% (ai ~bh)-1 &c,

(bh - eg) + 21 (eg - df) - 35 (df- e%

the law of which is evident. This shows also that these functions may be linearly expressed by means of the series of determinants a,

b

a,

b, c

b, c

b,

c, d

&c.

We may also immediately deduce from them the derivatives B which relate to two functions. For example, for functions of the sixth order this is ag' +a'g-6

(bf + b'f) + 15 (ce' + e'e) - 2Odd',

which has an obvious connection with ag - 66/+ 15ce - 10d2; and the same is the case for functions of any order. The following theorem is easily verified; but I am unacquainted with the general theory to which it belongs. "If U, V are any functions of the second order, and W=\U+ B2'[B2(W, W),

fiV;

then

B2(W,

(where B2' relates to X, fi) is the same that would be obtained by the elimination of x, y between Z7=0, F = 0." (See Note1.) In fact this becomes 4 {ac - 62) (oV - b") - (ac' + a'c - 2bbJ = 0, which is one of the forms under which the result of the elimination of the variables from two quadratic equations may be written. This is a result for which I am indebted to Mr Boole. 1

Not given with the present paper.

14]

ON LINEAR TRANSFORMATIONS.

101

Passing to the third degree, we may consider in particular the derivatives

nUVW=23a 31" 12a UVW=Ga(U, V, W): writing for shortness 2 r r A — ~~±*¥ \rY ' % *-»"£oyTJu — — TT' u ,

we have the general term Ca(U, V, W) = % {(-Y+s+t

ArAsAtU-*+t-sV-°-+r-tW'"+s-r},

where r, s, t extend from 0 to a. By changing the suffixes r, s the following more convenient formula

Ca(U, V, W) = 2 2 {(-)**U"V"W-*->"t

[(-)«Ap_tA,+t_aAt]},

where t extends from 0 to 2a: p, a-, and 3a — p — cr must be positive and not greater than 2a. In particular, according as a is odd or even, Ga(U, U, U) = 0, Ca(U, U, lT) =

QZZ{(-y+<'U'PU"'U^-<-''2[(-yAp_tAv+t_OiAt]\,

omitting therein those values of p, cr for which p > a or cr > 3a — p — cr, and dividing by two the terms in which p = cr or
B^iU, B,a(V, W)]; paying attention to the signification of B, this may be written

102

ON LINEAR TRANSFORMATIONS.

[14

where the symbols £<,, t\9 refer to the two systems x2> y2 : x3, ys. Thus it is easily seen that we may write , or 10=12 + 1 3 = 1 2 - 3 1 , whence the function becomes

(12 - 31T^t of which all the terms vanish except

Hence putting [4«]» 2 * 1 . 3 ... ( 4 a - 1 ) [2a]2*~ 2 . 4 . . . 4a

y

we have

Bia[U, B^(V, W)]=KCa(U, V, W),

or in particular U, U, U). Thus for example, neglecting a numerical factor, (ax2 + 2bxy + cy2) (c*2 + 2dxy + ey2) - (bx2 + 2cxy + dy2)2 = (ac -¥)xi + 2 (ad - be) a?y + (ae + 2bd - 3c2) xy + 2 (be- cd) xy3 + (ce - d2) y\ and then e {ac - b2) - id \ (ad- be) + Qc^(ae + 2bd- 3c2) - 4>b \ (be -cd)+a

(ce - d2)

= 3 (ace - ad? - b2e - c3 + 2bcd). We have likewise the singular equation

BUV, W)~*{^-^£:--. 1 where

U =T r i i o

+ r£)W

V, W)

/ f4al1 \ L a.af- - -j-J-a^-h) ...+ a^y** , &c.

If however £7= T = TT, we must write

the reason of which is easily seen.

This subject will be resumed in the sequel.

14]

103

ON LINEAR TRANSFORMATIONS.

The functions G may be transformed in the same way as the functions B have been. In fact

Ca(U, V, W)=T2a-k¥sa~kna~kGk(U,

V, W);

if in particular h = 1, then JJ.O JJ.l

C1(U, V,

ft,*

for U-\ &c.

but in general £ / %"ft0"%*ftT'%T <^i ( ^. F, W), where p + p' = a +
2a-2

2a

for [T'P, &c.

whence Ga(U, V, W) = SS {(-)"+• [/\p+2

where

/ =

L

r 1t

; < extends from 0 to a — 1; p,
each of them any positive values not greater than 2a — 2. In particular JJ,Sa-p-(r-2

Ga{U, U, U) = QXX {(->>+ f/-,p+2

where p, cr need only have such values t h a t p<
a,

9 -3 b, e, h c,

d,

f,

i

b,

c, g

f, 9

c,

d,

h

h

d,

e,

i

a,

e,

b, c,

g,

f

-3

+ 6 b,

d,

f

c,

e,

g

d, e, f

h

e

d, f,

-15

in which form it is obviously a linear function of the determinants

c,

d,

e

> f> 9

104

ON LINEAR TRANSFORMATIONS. a,

b,

c,

d,

e, f,

b,

c,

d,

e, f,

c,

d,

e, f,

g,

[14

g

g,

h

h,

i

which is true generally. Omitting for the present the theory of derivatives of the form 23" 31" 12V UVW, we pass on to the derivatives of the fourth degree, considering those forms in which all the differential coefficients are of the same order. We may write

UUVWX = (12. 34) (13.42) i(14 .23)yUVWX

= Ba^
v, W, Z) =

or if for shortness 12.34 = %I, we have

•D..A

1 3 . 42 = 33,

14.23 =

=a-im ^. uvwx.

Suppose U=V= W=X, and consider the derivatives which correspond to the same value / of a + fi + y. The question is to determine how many of these are independent, and to express the remaining ones in terms of these. Since the functions become equal after the differentiations, we are at liberty before the differentiations to interchange the symbolic numbers 1, 2, 3, 4 in any manner whatever. We have thus Da,fty = D?, y, a = Dy, „ „ = (~Y X»a, y, „ = ( - ) / Dy> ft a = ( - ) / Df, a, y J but the identical equation

a + 33 + © = 0, multiplied by ^,aW^c

and applied to the product UVWX, gives

whence if a + b +c =f— 1, we have a set of equations between the derivatives JDa^y for which a. + 8 + y=f. Reducing these by the conditions first found, suppose ®f is the number of divisions of an integer / into three parts, zero admissible, but permutations of the same three parts rejected. The number of derivatives is ©/", and the number of relations between them is ©(/—1). Hence ©/—©(/—1) of these derivatives are independent: only when f is even, one of these is Z ) / 0 0 , i.e. 12 34 . UVWX,

i.e. TifUV.WwX,

or Bf(U,

V) Bf (X, W), i.e. [Bf{U, U)J; rejecting this, the

number of independent derivatives, when /

is even, is (H)/—® (/"—1)—1.

be the greatest integer contained in the fraction ^ ; shown to be

Let E ij-

the number required may be

14]

ON LINEAR TRANSFORMATIONS.

105

according as / is even or odd. Giving to / the six forms 6g, 60 + 1, 6g + 2, 6^ + 3, 6^ + 4, 6g + 5, the corresponding numbers of the independent derivatives are 9> 9> 9> 9 + 1'

9> 9 + 1''

thus there is a single derivative for the orders 3, 5, 6, 7, 8, 10,... two for the orders 9, 11, 12, 13, 14, 16, ...&c. When / is even, the terms A--3,3,o, Dy_Si6J)..., and when / is odd, the terms A--i,i,o> A--4,4,o> A--7,7,o) &c- m a y be taken for independent derivatives: by stopping immediately before that in which the second suffix exceeds the first, the right number of terms is always obtained. Thus, when f=9 the independent derivatives are Aio, AM. and we have the system of equations = 0,

Dsa + Asi + A a ^ O ,

Al0 + A20 + All = 0,

DB40 + A1» + D441=0,

, + A 2 i = 0,

DB1 + D ifl + D « = OI

[ + Al2 = 0,

A22 + A32+A23 = 0, -^432 "T" -^342 T" -A-'333 = = " ,

X/630 ~H -^531 "l~ JJ&40 — " ,

which are to be reduced by = - A i o , &C.

It is easy to form the table Aoo = 0,

Aoo — -'-'410)

•^610,

A 2o = - A i o , Aoo=o,

An = 0,

A n = o,

J-'21O!

-^221

==

Dm = Aio,

0,

= o,

An = o,

D«

Aoo =

D» = 0, n : = 0,

n -'22O —

J

D211 = 0,

B?, 1 D] 2 -"4 ,

-^Bi

Daa = T) -1-'411 —

8 7) J •*'(

Dm , A 2 1 = ~ s A30 ~ ;

C.

14

106

ON LINEAR TRANSFORMATIONS. .B82,

Dm — -^710 = -^620 =

— -g •L'iSO H~

Au =

-^8101

(f -O8 )

-^720 =

|A 3«+1-B8 ,

-^SO) -D52I

=

Dm = 0,

2 -"8 >

2

~

J -^630 ~ T J -^8 ) 2

-0440= - i f A s o - ^ ^ ,

-^810!

An=0, -^AsO =

2

[14

-^Sa

=

2 -^810

2 -^'640>

i -^810 + 2 -^540>

DM0) -^531 = ~ 2 -^glO ~ \ -^540 >

A»=o, -D441 = 0, -Wm =

2 -^810 + 2 -L'640)

1*333=0.

Whatever be the value, all the tables except the three first commence thus, according as f is even or odd, -D/,0,0 =

Bf,

or Z)/i0>0 =0, /-2,

2,0=

but beyond this I am not acquainted with the law. To give some formulae for the transformation of these derivatives; we have, for example, Djr-i.i., = (12. S i / ' * 13. 42 UUUU = 13. 42 5 ^ (U, U) Bf_, (f7, V). But and

13.42 = fiT/2%^4 - ^ i ^ s ^ - ViV&Zt + Vi%£3
Bf.x (U; CT) £,_, (U, U) = B^ (f ff, 17F) 5 ^ (,,U, %U) = B , . , (CT.'CT. 1) Bf_, (U-*U-»), &c. 1

1

(where U>\ U- stand for U-'U- , &c); or

which reduces itself to

according as /" is even or odd.

14]

107

ON LINEAR TRANSFORMATIONS.

For example, for the orders 3, 5, 1, 9, we have Aio = - 2 {4 (ac - 62) (bd - c2) - (ad - 6c)2}, Dm = - 2 {4 (ae - 46d + 3c2) ( 6 / - 4ce + 3d2) - (af- 36e + 2cd)% Dm = - 2 {4 (ag - 66/+ 15ce - 10d2) (bh - &cg + 1 5 a / - 10e2) - (ah - 5bg + 9 c / Dm = - 2 {4 (ai - 8bh + 28c# - 56df+ 35e2) (bj - 8ci + 28dh - 56eg + 35/ 2 ) - (aj - Ibi + 20cA - 28a# + 14e/) 2 }. The derivatives D will be presently calculated in a completely expanded form up to the ninth order. We have, therefore, still to find the derivatives of the sixth and eighth orders, and a second derivative of the ninth order. For the sixth order, the simplest method is to make use of Dm, which is easily seen to be equal to 24

a,

6,

c, d

b,

c,

d, e

c,

d, e,

f

d,

e, f,

g

For the two others we have the general formulas

2

2

2

where U-°, U-1, U-2 have been written for U-\ U-\ Castrated in precisely t h e same way as t h a t for Df-i, i, 0 -

a formula which is demon-

5/_ 3 (U6Bf_s (U-

-2)Bf_s(U'*

- Bf_3 (U-°U<3) 5/_ 3 (U-s CT-•)j, ) £ , _ (U-2 o {in which U' °, &c. stand for U' °, &c). In particular 2

D^ = 2 {4 (a# - 66/ + 15ce - lOd2) (ci - Uh + 15eg- Wf) - 4 (ah - hbg + 9c/ - bde) x (6i + 5ch + 9dg - oef) + (ai - 6bh + lQcg - 26df+ 15e2f +8(bh-

6cg + lodf-

10e2)2},

Dm = _ 2 {4 (a# - 6 6 / + 1 5 c e - 10rf2) (dj-Qei + 15/A - 10^2) -6 (ah- hbg + 9c/-5cfe) x (cj- hdi + 9e/i - 5fg) + 6 (at - 66A + 16c^ - 2Qdf+ 15
108 Ai» = 2 x a?d2 abed ac3 b3d b2c2

4 1 -6 44 44 - 3 + 9

2 2

[14

ON LINEAR TRANSFORMATIONS.

a / 4 abef — acdf+ ace2 4 ad2e b*df + b2e? + bc2f bede — bds 4 cse 4 c2d2 -

1 10 4 16 12 16 9 12 76 48 48 32

A - = 24 x 1 1 1 2 ae3 - 1 b2eg - 1 b2f2 4-1 bedg 4 2 beef - 2 bd'f- 2 bde2 4 2 c3g - 1 c2df 4 2 c2e2 4- 1 cd2e - 3 d* 4 1

aceg + acf2ad2gadef+

+ 12

± 142

,= 2 x a2h2 4 abgh— acfh + acg2 4 adeh— adfga&g 4 b2fh + by +

1 14 18 24 10 60 40 24 25 bceh — 60 befg - 234 bd% + 40 bdeg + 50 bdf2 +360 b*f- 240 c2eg 4 360 c 2 / 2 + 81 ed2g- 240 cdef — 990 ce3 + 600 d3f + 600 d2e2 4 375 2223

A« = aH* abhi acgi acti1 adfi adgh ae2i aefh aeg2 af2g b2gi b2h2

4

2x 1 16 36 20 52 60 30 20 60 40 20 44 60 388 20 696 180 340 460 240 60 180 544 40 460 2596 2340 420 240 420 12876 3280 1025

4 4

4 4 4 4 4

befi begh bdei bdfh bdg2 be2h befg

4 4 4 bf3 4 c2ei 4 c2fh 4 cy2 4 cd i cdeh cdfg c
± 8632

Dm = 4 -D222 4 i B£ > J) — — 3 J) _|-.12?2 n 530 o r>62° r>8 '

equations which give the functions ^330> -^530> -D64o> b y m e a n s of which ^ e o t ^ e r s ^ a v e been expressed.

A !=4x

D 8 M == 2 x aV 4 abii — achj + aci2 + adgjadhi —

1 18 40 32 56 112 28 224

o-efj + aegi + aeh2 af2i 140 afah ags b2hj + 32 b2i2 4 49 112 begj bchi — 536 bdfj + 224 392 bdgi 4 bdh2 + 896 be2j 140 befi — 196 begh — 1792 bf% + 1120 bfq2

+ — 4 4 +

-f.

4 4 4 4 4

J iJ

dfj 2

c gi 4 c2A2

4

cdej cdfi — cdgh— ceH + cefh + ceg* 4 cf2g 3 d j \M J

896 400

4 4

1792 4256 1120 560 6272 3920

d2fh + 6272 784 dy 4 d?ei .. , , , de2h - 3920 defg- 13328 df3 4 7840 e>g 4- 7840 e*f2 - 4704 + 35022

4

2 2 7 7 5 22 27 25 45 20 2 2 7 7 22 74 52 25 73 23 70 45 27 52 25 45 23 22 70 127 32 25 20

• 32

4• 4 7 4- 45 • 25

44-

85 50 50 30

i : 698

14]

ON LINEAR TRANSFORMATIONS.

109

We may now proceed to demonstrate an important property of the derivatives of the fourth degree, analogous to the one which exists for the third degree. Let U, V, F , X be functions of any order f: then, investigating the value of the expression B^[Ba{U, V), Ba(W,X)], this reduces itself in the first place to

where £9, rje refer to U and V, and £j,, T?^ to F and X: this comes to writing Vi', whence 00 = 13+14 + 23 + 24, or the function in question is (13 + 14 + 23 + 24)^~2a 12" 34* UVWX. But all the terms of this where the sum of the indices of £ , % or ^2, rj2 or ?3> Vs o r ?4» Vi> exceed f, vanish: whence it is only necessary to consider those of the form Kr (13 .42) r (14 . 23/~ a " r (T2.34)° UVWX, where Kr denotes the numerical coefficient [r]r [rj [ / - a - r]/-«-- [ / - a - r ] ^ - ' ' or

5 2/ _ 2a [ 5 . (0; F), 5 a ( F , X)] = S {iTrI>a, r , y_a_r ( C7, F, F , X)}.

In particular, if 1/"= F = F = X , writing also S a for Ba(U, IT), B^(Ba,

Ba) = % (Kr Da, r , / _ a _ r ).

If a is odd, this becomes an equation which must be satisfied identically by the relations that exist between the quantities D: if, on the contrary, a is even, we see that there are as many independent functions of the form as there are of the form D; and that these two systems may be linearly expressed, either by means of the other. Thus, for the orders 3, 5, 7, the derivatives D are respectively equal, neglecting a numerical factor, to for the sixth order they may be linearly expressed by means of Bn{U\

IP), Bt;

and so on. All that remains to complete the theory of the fourth degree is to find the general solution of this system of equations, as also of the system connecting the derivatives B.

110

fi.fs

ON LINEAR TRANSFORMATIONS.

[14

Passing on to a more general property; let TJlt £72, ... Up be functions of the orders •••fP; and suppose

a function of the degree fx in the variables: suppose that © (JJ%... Up) contains the differential coefficients of the order r2 for U^, rs for Us, &c, so thaA,f1 = (f2—r2)+...(fp—rp). Consider the expression

BfAUlt which reduces itself in the first place to ( 1 2 + 1 3 . . , + l p f ' a t / ! ^ . . . Up> K (12 2 *"2 13 s ~*\.. l p " r" • Ux ?72... trp ;

then to where for shortness

For if one of the indices were smaller another would be greater, for instance that of 12: and the symbols f2, ??2 in 12 22 • the term would vanish. Hence, writing

and

8^(^,17,...,

we have

would rise to an order higher than / 2 , or

Up) ^U^.U,

...Vp,

BA {Ult ® (Ua, ... Up)} = K& {U,, U2... Up);

i.e. the first side is a constant derivative of Ult U^.-.TIp. Ux = f-j^r (a^1 + ...), L/J 1

Suppose

iU,... Up) = a0Afl - & a.Af^

then

i.e. or

+...;

i ^ z i - e V ^ . . . ^ , ^ ^ =^0^(17,, finally,

0(0,, ... Up) = ^

[ ^

-x^y

^

U2...UP)...,

+ ...) & ( ^ ... U,).

an equation which holds good (changing, however, the numerical factor,) when several of the functions U1... Up become identical. Hence the theorem: if U be a function given by

14]

ON LINEAR TRANSFORMATIONS.

Ill

and © be any constant derivative whatever of U, then (xf -, xf-^ii -= h ...) ® V da/ da/-! / is a derivative of U, and its value, neglecting a numerical factor, may be found by omitting in the symbol • , which corresponds to the derivative ®, the factors which contain any one, no matter which, of the symbolic numbers. If, for example,

or

then

(^i-^y-^ + ^i-y^y

V dd

dc

y

db

y

da

reduces itself, omitting a numerical factor, to 122 13 UUU= - \B, {U, B, {U, U)}. This may be compared with some formulae of M. Eisenstein's (Grelle, vol. xxvu. [1844, pp. 105, 106]; adopting his notation, we have $ = ax3 + 3bx2y + Zcocf + dtf, F = ^ B2 ( $ , 3>) = (ac - 62) a? + (ad - be) xy + (bd - c2) y%,

^ = -I 2 (a? ^ _ a*y* +xf^y y V da

dc

db

y

fP) D, da)

where D is the same as ©. Hence to the system of formulae which he has given, we may add the two following:

*~

*{dx dy

dy dx)' d

( dx* dy

da?dy\

® dy

dxdy

dxdy2 \ dxdy

dx

d

® d

dx* dy)

dy3

dx2 dx) '

the first of which explains most simply the origin of the function <£>!. It will be sufficient to indicate the reductions which may be applied to derivatives of the form Catfi7(U,

V, W)

where II, V, W are homogeneous functions.

In fact, if

& + vy= S, the above becomes, neglecting a numerical factor, (a. W

• (B, • Slf • (B,. 12)v UVW,

112

[14

ON LINEAR TRANSFORMATIONS.

where the symbols £, rj are supposed not to affect the x, y which enter into the expressions S- But we have identically

an equation which gives rise to reductions similar to those which have been found for the derivatives Z)a)/3]Y, but which require to be performed with care, in order to avoid inacccuracies with respect to the numerical factors. It may, however, be at once inferred, that the number of independent derivatives (7a>/3j y is the same with that of the independent derivatives Da< & y for the same value of a + /3 + j . From similar reasonings to those by which B {U, B{U, U)} has been found, the following general theorem may be inferred. " The derivative of any number of the derivatives of one or more functions, or even of any number of functions of these derivatives, is itself a derivative of the original functions." For the complete reduction of these double derivatives, it would be sufficient, theoretically, to be able to reduce to the smallest number possible, the derivatives of any given degree whatever. This has been done for the derivatives of the third degree Ca, p, y, a n ( i f° r those of the fourth degree, in which all the differentiations rise to the same order (D aj|3]7 ): it seems, however, very difficult to extend these methods even to the next simplest cases,—extensive researches in the theory of the division of numbers would probably be necessary. Important results might be obtained by connecting the theory of hyperdeterminants with that of elimination, but I have not yet arrived at anything satisfactory upon this subject. I shall conclude with the remark, that it is very easy to find a series, or rather a series of series of hyperdeterminants of all degrees, viz. the determinants a,

b

a,

b,

c

a, b,

b,

c

b,

c,

d

6,

c,

d, e

a,

26,

c

a,

46, 6 c , 4c?, e

b,

2c, d

b,

4 c , 6c?, 4e, /

a,

26,

c,

b,

2c,

d,

c, 4>d, 6e,

&c.

c, d, e

c, d, d,

c, d e,

f

e, f,

g

&c.

4
a,

36,

3c,

d

b,

Be,

3d,

e

a,

36,

3c,

d,

.

6,

3c,

3d,

a,

46, 6 c , 4c?,

e,

b,

4c, 6c?, 4 e ,

/,

.

a,

36,

3c,

d,

c,

4d,

g,

.

6,

3c,

3c?,

e,

6e, 4/,

&c.

[I have inserted in these determinants the numerical coefficients which were by mistake omitted.] However, these functions are not all independent; e.g. the last may be linearly expressed by the square of the second and the cube of (ae — %d + 3c2); nor do I know the symbolical form of these hyperdeterminant determinants.

15]

113

15. NOTE SUR DEUX FORMULES DONNEES PAR M. M. EISENSTEIN ET HESSE. [From the Journal fur die reine und angewandte Mathematik, (Crelle) vol. XXIX. (1845), pp. 54—57.] a donne [Journal, t. xxvu. (1844), pp. 105—106] cette formule assez

ME. EISENSTEIN

remarquable : (a2da - 362c2 + 4ac3 + 42- 3B>C* + bAO* + 4DJ53 - 6ABCD, ou A, B, G, D sont des fonctions donndes de a, b, c, d. Cela peut se gdn^raliser comme suit. Soit u = a?h? + by + c 2 / 2 + d2e2 - lahbg - 2ahcf- 2ahde - 2bgcf- 2bgde - 2cfde + kadfg + 4
, du da

2

„

, du db

2

„

. du dc

2

TT

, du dh

Repre'sentons par U la meme fonction de A, B, C, ... H, que la fonction u Test des quantity's a, b, c, ... h, Ton a l'dquation

C'est un cas particulier d'une propriety g^nerale de la fonction u, que Ton peut e'noncer ainsi. Imaginons la fonction dx1yiz2 +'ex2y1z

transforme'e en a'x\y\z\ C.

+ b'x\y\z\

+ ...

+

h'x'^y'^, 15

114

NOTE SUE, DEUX FORMULES DONNEES PAR

[15

par les substitutions 2,

yx = fi1y1'4-/42 yi,

z1 =

v1z1'+v2z,',

et soit u', la fonction analogue a u, des nouveaux coefficients a', V, c', ... h': alors u' = (\ V - X2X/)2 fa V2' - ^ f^J {yx v2 - v2 Viy . u. En echangeant seulement les z, ceci donne

ou

a' = v1 a + v^e, b' — v1b + v^f, e' = v2a + v2'e, f = v2b + v2f,

Soit

c' = vx c + v/^f, g' = v2c+ v2g,

d' = ^ d + v/h, h! =v2d + v2'h.

vx =ah — bg — cf— de, Vi=-2

(ad - be),

v2=-2

(eh -fg),

v2 =ah — bg — cf-de

(= vx),

on trouve d'abord (viv2 — v2v2) = u?, ou u' = u3,

et puis, en ayant e"gard aux valeurs de A, B, C, ... H:

a'=H, e'=B,

V = -O, f' = -B,

c' = -F, g' = -G,

d' = E, h'=A,

de maniere que v! = U, d'ou enfin U = u\ La propriete de la fonction u que je viens d'dnoncer, se rapporte a une theorie assez g4n4isb\e, d'une nouvelle espece de fonctions alge"briques dont je m'occupe actuellement, et lesquelles a cause de leur analogie avec les de"terminantes, on pourrait comme je crois nommer " Hyperdeterminantes." Je me propose de poser les premiers fondemens de cette the'orie dans un memoire qui va paraitre dans le prochain No. du Cambridge Mathematical Journal (No. XXIII.) [13]. A present je passe a une formule de Mr. Hesse [Journal, t. XXVIII. (1844), p. 88], qui se rapporte aussi a la meme the'orie. Soit V, une fonction homogene du troisieme ordre, et a trois variables *, y, z. Soit W, la ddterminante forme'e avec les coefficients du second ordre de V, arranges en cette forme: da? '

dxdy'

dxdz '

dxdy'

dy* '

dydz'

dzdx'

d*V dzdy'

d?V

15]

M. M. EISENSTEIN ET HESSE.

115

soit a une quantite constante quelconque et soient A et B deux autres constantes de'termine's: Mr. Hesse a de'montre' liquation remarquable

mais sans donner la forme des coefficients A et B, ce qui parait &bre tres difficile a effectuer. En consideYant le cas d'une fonction homogene de deux variables seulement, mais d'ailleurs d'un ordre quelconque, on parvient a un theoreme analogue qui m'a para interessant. " Soit U une fonction homogene et de l'ordre v des deux variables x, y, et V U la de"terminante ( i > - 2 ) 0 - 3 ) 3 . V(U + aVU)={-v(v-l)(v-3y>Ja+v{v-l)(2v-5fIa?} U 3 + {(v-2)(v- 3) + (v - 2) {v - 3) (2v - 5) Ja?} V {7. En repre"sentant par i, j , k, I, m, les coefficients differentiels du quatrieme ordre de U, on a / = ikm - iP -jm? - k% + 2jkl, J = ijl -3k2- mi, de maniere que /, J sont des fonctions de *, y des ordres 3 (v — 4) et 2 (v — 4>) respecti vement." Ce qu'il y a de remarquable, c'est la forme de ces deux quantites /, J. On voit d'abord que la fonction J est la determinante forme'e avec les termes i, j , k j , k, I k, I, m. Mais de plus les deux fonctions /, J ont la propriety suivante. Si Ton imagine une fonction du quatrieme ordre

transform^e en

fV 2 au moyen de

en repr^sentant par /',
J' = (V-x'fiy.J,

T = (V-\»4./. 15—2

116

NOTE SUE, DEUX FORMULES DONNEES PAR M. M. EISENSTEIN ET HESSE.

[15

L'on parviendrait, je crois, a des rdsultats plus simples en considerant une fonction U, homogene en x, y, et aussi en xx, ylt et en posant dxdxx' Par exemple, si U est du second ordre en x, y et aussi en xlt yu de maniere que U=

x? (A a? + 2B xy + G y2) + 2X& (A' x' + 2B'xy+C' y*) + yi* (A"x*+2B"xy+C"f);

on a simplement en reprdsentant par @L la determinante formee avec les coefficients A , B, G; A', B, C; A", B", C"; et multipliee par 28. Mais il faudrait d^velopper tout cela beaucoup plus loin. P.S. Je ne sais pas si Ton a jamais disrate" la question suivante, qui doit conduire, il me semble, a des resultats importants. Soient L, M, L', M', U, V des fonctions de x. En ^liminant cette variable entre les Equations LU+ MV=0 et L'U+ M'V= 0, et en reprdsentant liquation ainsi obtenue par [LU + MV, L'U + M'V] = 0, et de m§me par [U, F] = 0 Y^quation obtenue en eliminant x entre U = 0, V=0, il est clair que [LU + MU, L'U + M'V~\ contiendra [U, V] comme facteur: quelles sont maintenant les proprie"t6s de l'autre facteur qu'on peut ecrire sous les trois formes: [LU+MV, L'U + M'V] : [U, V], [LU + MV, LM' - L'M] : [L, M] et [L'U + M'V, LM'-L'M] : [L', M']) ?

16]

117

16. MEMOIRE SUR LES HYPERDETERMINANTS (TRADUCTION D'UN MEMOIRE ANGLAIS INSERE DANS LE "CAMBRIDGE MATHEMATICAL JOURNAL" AVEC QUELQUES ADDITIONS DE L'AUTEUR). [From the Journal fur

die reine und angewandte Mathematik (Crelle), vol. XXX. (1846), pp. 1 - 3 7 . ]

This is a translation of the foregoing papers, [13] and [14], On Linear Transformations, the additions are inconsiderable: in date of publication it was subsequent to [13] but I think preceded [14].

118

[17

17. NOTE ON MR BRONWIN'S PAPER ON ELLIPTIC INTEGRALS. [From the Cambridge Mathematical Journal, vol. in. (1843), pp. 197—198.] This Note was in answer to objections raised by Mr Bronwin in his paper "On Elliptic Functions" Camb. Math. Jour. vol. in. (1843) pp. 123—131, to some of the formulas of transformation in the Fundamenta Nova. There is in it a serious error which was afterwards pointed out by Mr Bronwin. The Note is omitted, as are also the other papers, [18] and part of [21], which refer to the same subject.

18]

119

18. REMARKS ON THE REV. B. BRONWIN'S PAPER ON JACOBI'S THEORY OF ELLIPTIC FUNCTIONS. [From the Philosophical Magazine, vol. xxil. (1843), pp. 358—368.] This paper is omitted: see [17].

120

[19

19. INVESTIGATION OF THE TRANSFORMATION OF CERTAIN ELLIPTIC FUNCTIONS. [From the Philosophical Magazine, vol. xxv. (1844), pp. 352—354] THE function sinam u ((ftw for shortness) may be expressed in the form + 2mK +2m'K'i) *

W

where m, m! receive any integer, positive or negative, values whatever, omitting only the combination m = 0, m'= 0 in the numerator (Abel, (Euvres, t. I. p. 212, [Ed. 2, p. 343] but with modifications to adapt it to Jacobi's notation; also the positive and negative values of m, m! are not collected together as in Abel's formulae). We deduce from this _ / u ) • u(l \ ^ (2) ~ V 2mK + 2m'K'i + e) ' \ 2mK + (2m' + 1)K'%+ 8J'"{ h Suppose now K = aH + a'H'i, K'i = bH + b'H'i, a, b, a', b' integers, and ab' — a'b a positive number v. Also let 6 =/H +f'H'i; f,f integers such that af'—a'f, bf'—b'f, v, have not all three any common factor. Consider the expression ... (ft (u + 2 (y — 1) to) U ^^

from which

j n (1 +

j ( 4 )

where r extends from 0 to v — 1 inclusively, the single combination m = 0, m = 0, r = 0 being omitted in the numerator. We may write

mK + m'K'i

19] INVESTIGATION OF THE TRANSFORMATION OF CERTAIN ELLIPTIC FUNCTIONS. 121 (i, /A denoting any integers whatever. Also to given values of fi, fi there corresponds only a single system of values of m, m', r. To prove this we must show that the equations ma + m'b + rf = p,

can always be satisfied, and satisfied in a single manner only. Observing the value of v, vm + r(Vf-bf')=fib'-plb; then if v and b'f—bf have no common factor, there is a single value of r less than v, which gives an integer value for m. This being the case, m'b and m'b' are both integers, and therefore, since b, b' have no common factor (for such a factor would divide v and b'f— bf), m! is also an integer. If, however, v and b'f— bf have a common factor c, so that v—ab'— a'b = c, b'f— bf = c'; then (af — a'f) V = c (<£/' — '/), or since no factor of c divides af — a'f c divides b', and consequently b. The equation for v may therefore be divided by c. Hence, putting - = v,, we may find a value of c r, say r,, less than vt, which makes m an integer; and the general value of r less than v which makes ,m an integer, is r =r/+sv/, where s is a positive integer less than c. But m being integral, bm', b'm', and consequently cm' are integral; we have also cvxm' + (rl + sv) {af — a'f) = a/i' - a'/tt; and there may be found a single value of s less than c, giving an integer value for m'. Hence in every case there is a single system of values of m, m', r, corresponding to any assumed integer values whatever of fi, /A'. Hence = UU l + o „

,

U-II l + s - g

~

^-rrr. )=6,U

(5)

4>,u being a function similar to u, or sin am u, but to a different modulus, viz. such that the complete functions are H, H' instead of K, K'. We have therefore 2a>)... 0 (u + 2 (v- 1) to)

{v - 1) co)

[)M

Expressing w in terms of K, K', we have vH = b'K — a'K'i, — vH'i = bK — aK'i, and

therefore v(o — (b'f—bf')K—(a'f—af')K'i. Let g, g' be any two integer numbers having no common factor, which is also a factor of v, we may always determine a, b, a', V, so that vco=gK- g'K'i. This will be the case if g = b'f- bf, g' = a'f- af. One of the quantities f f may be assumed equal to 0. Suppose / ' = 0, then g = b'f g' = a'f; whence ag—bg'=vf. Let k be the greatest common measure of g, g', so that g — hgt, g' = kg'/, then, since no factor of k divides v, k must divide /, or f=kf, but g, = b'f g', = af, and a', b' are integers, or f must divide g, g'; whence fl = 1, or f= k. Also agt — bg't = v, where g/ and g'l are prime to each other, so that integer values may always be found for a and b; so that in the equation (1) we have (7), g, g' being any integer numbers such that no common factor of g, g' also divides v. C 16

122 INVESTIGATION OF THE TRANSFORMATION OF CERTAIN ELLIPTIC FUNCTIONS. [19

The above supposition, f = 0, is, however, only a particular one; omitting it, the conditions to be satisfied by a, b, a', V, may be written under the form ab' - a'b = v,

)

ag -bg' = 0 [mod. v], >

a'g-b'g'=0[xaod.

(8)

v],j

to which we may join the equations before obtained, vH=b'K-a'K'i, vH'i=bK-aKi, ' which contain the theory of the modular equation. This, however, involves some further investigations, which are not sufficiently connected with the present subject to be attempted here.

20]

123

20. ON CEETAIN BESULTS RELATING TO QUATERNIONS. [From the Philosophical Magazine, vol. xxvi. (1845), pp. 141—145.] IN his last paper on Quaternions [Phil. Mag. vol. xxv. (1844), p. 491] Sir William R. Hamilton has alluded to a paper of mine on the Analytical Geometry of («) Dimensions, in the Cambridge Mathematical Journal [11], as one that might refer to the same subject. It may perhaps be as well to notice that the investigations there contained have no reference whatever to Sir William Hamilton's very beautiful theory; a more correct title for them would have been, a Generalization of the Analysis which occurs in ordinary Analytical Geometry. I take this opportunity of communicating one or two results relating to quaternions ; the first of them does appear to me rather a curious one. Observing that it is easy to form the equation (A + Bi -h Cj + Die)-1 (a + /3i+ yj + 8k) (A + Bi + Cj + Dk) 1

I +i[ /3 {A2 + B*-C*-D*) + 2y(BC+AD) I +j [2/3(BC -AD) ( + k[20{BD + AC)

+28{BD-A(J)

]

+ y(A* -B* + C*-D*) + 28{CD + AB) +2y(CD-AB)

] 2

+8(A* - B*- O + D )] J

which I have given with these letters for the sake of reference; it will be convenient to change the notation and write 16—2

124

[20

ON CERTAIN RESULTS RELATING TO QUATERNIONS.

(1 + Xi + fij + vk)-1 (ix +jy + kz) (1 + Xi + pj + vk) 1 ~ 1 + X2 + ju.2 + z>2

i [ x (1 + X2 - fj? - 1

2y

2z(Xv-li)

,... (3)

-j [2x (X/J, - v) + k \2x (Xv + fi)

=

i (ax + a'y + a"z)" (4)

suppose. The peculiarity of this formula is, that the coefficients a, /3,... are precisely such that a system of formulas xt = ax + a'y + a"z ~)

y, = ffx +ft'y+ $"z > z, = yx + y'y + y"z j denotes the rectangular. Suppose the by means of at angles f,

(5)

transformation from one set of rectangular axes to another set, also Nor is this all, the quantities X, ytt, v may be geometrically interpreted. axes Ax, Ay, Az could be made to coincide with the axes Axt, Ayn Azl a revolution through an angle 8 round an axis AP inclined to Ax, Ay, Az, g, h then X = tan ^0 cosy,

= tan

v — tan \6 cos h.

In fact the formulas are precisely those given for such a transformation by M. Olinde Rodrigues Liouville, t. v., " Des lois geometriques qui rdgissent les de"placemens d'un systeme solide" (or Camb. Math. Journal, t. iii. p. 224 [6]). It would be an interesting question to account, a priori, for the appearance of these coefficients here. The ordinary definition of a determinant naturally leads to that of a quaternion determinant. We have, for instance, (6),

•B7 .

•sr',

> 4> • x = ",

4>", y "

&c, the same as for common determinants, only here the order of the factors on each term of the second side of the equation is essential, and not, as in the other case, arbitrary. Thus, for instance, •37,

'CJ

— OTtiT — •OTOT ,

= 0

(7),

20]

125

ON CERTAIN RESULTS RELATING TO QUATERNIONS. •m,

tss

=

but

TJSTS1 — TS'TB,

4=

0

•(8),

that is, a quaternion determinant does not vanish when two vertical rows become identical. One is immediately led to inquire what the value of such determinants is. Suppose OT = x + iy +jz + lew, a' = x' + iy' +jz' + kw', &c, it is easy to prove •sr

•07

= - 2 i , j , lc

(9),

x', y', z' '57 ,

137 ,

"37

= -2 3 , * , j , k

x , y, x', y',

z , w z1, w'

x",

z", • w"

y",

= 0

/// 7

,

/// 1S7

/// ,

"37

(10),

(11).

/// ,

'C7

or a quaternion determinant vanishes when four or more of its vertical rows become identical. Again, it is immediately seen that (12)

+

0', a-' &c. for determinants of any order, whence the theorem, if any four (or more) adjacent vertical columns of a quaternion determinant be transposed in every possible manner, the sum of all these determinants vanishes, which is a much less simple property than the one which exists for the horizontal rows, viz. the same that in ordinary determinants exists for the horizontal or vertical rows indifferently. It is important to remark that the equations r, I.e.

<£ = 0 or

vr, w' =0, &c

(13)

vrd}' — ur'tb = 0," or urd)' — (bus — 0, &C.

are none of them the result of the elimination of II, , from the two equations •
' I I + <£' = 0,

(14).

126

ON CERTAIN RESULTS RELATING TO QUATERNIONS.

[20

On the contrary, the result of this elimination is the very different equation OT-1 . -OT'-1.(/>' = 0

(15),

equivalent of course to four independent equations, one of which may evidently be replaced by 0

(16),

if MTZ, &C. denotes the modulus of vs, &c. An equation analogous to this last will undoubtedly hold for any number of equations, but it is difficult to say what is the equation analogous to the one immediately preceding this, in the case of a greater number of equations, or rather, it is difficult to give the result in a symmetrical form independent of extraneous factors. I may just, in conclusion, mention what appears to me a possible application of Sir William Hamilton's interesting discovery. In the same way that the circular functions depend on infinite products, such as *l(l +

) , &c

(17),

{m any integer from oo to — oo, omittingTO= 0} and the inverse elliptic functions on the doubly infinite products ;(l+

5

. ) , &c

(18)

V TOW + *—" '

[m and n integers from oo to — oo , omittingTO= 0, n = 0}, may not the inverse ultra-elliptic functions of the next order of complexity depend on the quadruply infinite products + A—r. ry)? (19) V mw + nun + oqyj -\-p-^rkj {TO, n, o, p integers from oo to — oo , omitting m = 0, n = 0, o = 0, p = 0}. XYL(\

It seems as if some supposition of this kind would remove a difficulty started by Jacobi (Orelle, t. ix.) with respect to the multiple periodicity of these functions. Of course this must remain a mere suggestion until the theory of quaternions is very much more developed than it is at present; in particular the theory of quaternion exponentials would have to be developed, for even in a product, such as (18), there is a certain singular exponential factor running through the theory, as appears from some formulae in Jacobi's Fund. Nova (relative to his functions ©, H), the occurrence of which may be accounted for, & priori, as I have succeeded in doing in a paper to be published shortly in the Cambridge Mathematical Journal [24].

21]

127

21. ON JACOBI'S ELLIPTIC FUNCTIONS, IN REPLY TO THE REV. B. BRONWIN; AND ON QUATERNIONS. [From the Philosophical Magazine, vol. xxvi. (1845), pp. 208, 211.] The first part of this Paper is omitted, see [17]: only the Postscript on Quaternions, pp. 210, 211, is printed.

IT is possible to form an analogous theory with seven imaginary roots of (— 1) (? with v = 2n — 1 roots when v is a prime number). Thus if these be ilt iit is, it, ie, ie, i7, which group together according to the types 123,

145,

624,

653, 725,

734,

176,

i.e. the type 123 denotes the system of equations O 1 fc

^^

ft

l*2 —

&

^2^1

=

0 1

3J

^3J

^—

t v

23 —

^3^2

=

4

rt />

vi)

b

^11

^1^3

$v\

^—

— =

n

V2) ~™ ^ 2 >

&c. We have the following expression for the product of two factors:

L 0 — JL !X.!

where

[23 [31

Hh H-

[12

H-

— J L 2 A . 2 . . . — 2£7X i

45 -h 46 Hh 65 Hh

76 H57 Hh 47 Hh

(01)] h (02)] s

[51 Hh

62

Hh [24 -h

36 Hh

47 Hh 72 -1-

(04)] i<

[14

[25

34

-1-

53

Hh

-1-f-

(03)] i

(05)] i, 17 Hh (06)] i. 61 -H (07)] i

(01 ) = XOZ'X + Z j Z ' o . . . ; 12 = X1X'3 - XJd

&c.;

and the modulus of this expression is the product of the moduli of the factors. The above system of types requires some care in writing down, and not only with respect to the combinations of the letters, but also to their order; it would be vitiated, e.g. by writing 716 instead of 176. A theorem analogous to that which I gave before, for quaternions, is the following:—If A = 1 +Xjij ... +X,i7, X — xfa ... +x7i7: it is immediately \,...i7 shown that the possible part of A - 1 XA vanishes, and that the coefficients of are linear functions of xu...x7. The modulus of the above expression is evidently the modulus of X; hence " we may determine seven linear functions of xx... w7, the sum of whose squares is equal to $•?+... + ay*." The number of arbitrary quantities is however only seven, instead of twenty-one, as it should be.

128

[22

22. ON ALGEBEAICAL COUPLES. [From the Philosophical Magazine, vol. xxvu. (1845), pp. 38—40.] IT is worth while, in connection with the theory of quaternions and the researches of Mr Graves {Phil. Mag. [Vol. XXVI. (1845), pp. 315—320]), to investigate the properties of a couple ix +jy in which i, j are symbols such that v> = ai +Sj, ij=a'i + §'j, ji =ji+ Bj, If then

ix +jy ixx + jyx = iX + j Y, X = a xx1 + a'xy1 +
Imagine the constants a, S... so determined that ix +jy may have a modulus of the form K {x + \y) {x + fiy) ; there results one of the four following essentially independent systems A.

i2 = ^- {SX/i + Xfl

f = - XpSi + (7 + X + X + XY = J (7 + XB) {x + Xy) {x, + Xy,), X + /JLY= - (7 + fM$) {x + fiy) {x + fiyj.

22]

ON ALGEBRAICAL COUPLES.

129

The couple may be said to have the two linear moduli, hry + XB)(x + Xy), A»

fit

as well as the quadratic one, ~ (7 + XB) (7 + fiB) (x + Xy) (x + fiy), the product of these, which is the modulus, and the only modulus in the remaining systems. B.

tf

= - Bi + —(7 +

BiTJi)j,

h/jJb

f = (X + y, 7 + XfiB) % - 7J, I X + XY= - (7 + XB) (x + fiy) (x, + ^Vl), 1 [ X + pY= i (7 + fiB) (x + Xy) (x, + Xy,).

a

i-'L^

+ yx +

fti-j,

+X*

' X+ XY= -(7 + XS)(x + Xy) (x, + fiy1), [X+ tiY= - (7 + pB) (x + py) (xx + Xy,).

D,

% — — 0% -\- —

X" +

j 2 = - XyitSi + (7 + X + fiB)j, I X + XF= - (7 + X8) (x + fiy) (x, + Xyt), ~(y + fiB)(x + Xy) (x,

17

130

ON ALGEBRAICAL COUPLES.

[22

The formulae are much simpler and not essentially less general, if fi = — X. They thus become A'.

*=

Si

+lj,

ij = j i = y{

+ Bj,

2

j=

X*8i + yj,

X ± XY= ± J ( 7 + XB) (x ± Xy) (x, ± XyJ. (Two linear moduli.) "R'

V2 — — &i

— %- i

V =fi = 7* +

S

f = - \*8i -

yj,

J>

X + XY=+l(y±XB)(x

+ Xy) (xx + \ y i ) .

ji = T» + Sj, / = - X « - 7J, Z + XF= ± ^ (7 + XS)0 + Xy) (x, + Xy,). D'. ij = -yi

-

Sj,

X ± XY= +hy ±XB)(x + Xy) (x, ± Xy). There is a system more general than (A.) having a single linear modulus q(0X + Y): this is E. i1 = a (i - 6j) + 6>qj,

qj 0X+Y=q(6 or, without real loss of generality,

22]

ON ALGEBRAICAL COUPLES.

E'.

131

i* = a. i, ij = a'i, ji = y i, f = y'i + qj,

To complete the theory of this system, one may add the identical equation 1 ,

where

TT

Q (& - Ma) I ,.

a' - 0-y

6 (y - By') - (a - Oa!)

Jkf = — ^

f^—^ ay— ay

'•

By determining the constants, so that -^ -,, = ~ = ~ , the system would J & J d-qM a-da a — 6y reduce itself to the form A.

17—2

132

[23

23. ON THE TRANSFOEMATION OF ELLIPTIC FUNCTIONS. [From the Philosophical Magazine and Journal of Science, vol.

XXVII.

(1845), pp. 424—427.]

IN a former paper [19] I gave a proof of Jacobi's theorem, which I suggested [21] would lead to the resolution of the very important problem of finding the relation between the complete functions. This is in fact effected by the formulas there given, but there is an apparent indetermmateness in them, the cause of which it is necessary to explain, and which I shall now show to be inherent in the problem. For the sake of supplying an omission, for the detection of which I am indebted to Mr Bronwin, I will first recapitulate the steps of the demonstration. If £ &), | v (x) be the complete functions corresponding to <£, x, then this function is expressible in the form

i+—5—Un(i+—^ ma> + nv/ \ Let p be any prime number, fi, v integers not divisible by p, and a _

/.&> + vv P

The function

is always reducible to the form

V

mm +nvj 1

'

Analogous to the K, K'i of M. Jacobi.

23]

ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.

133

and thus fax is an inverse function, the complete functions of which are \ to', \ v : and to', v are connected with to, v by the equations to' = ~ (am + §v), v' = ~(a'to + §'v), P a, §, a', §' being any integers subject to the conditions that a, §' are odd and a', § even ; also a§'-a'S = p, fi§' — vaf = I'p, fi§ — va = Ip, I, I' being any integers whatever. In fact, to prove this, we have only to consider the general form of a factor in the numerator of tf-jx. Omitting a constant factor, this is

and it is to be shown that we can always satisfy the equation TOCO + nv + 2rd = m'u>' +

n'v,

or the equations pm + 2rfi = m'a. + n'a', pn+ 2rv =m'g + n'S'; and also that to each set of values of TO, n, r, there is a unique set of values of TO', ri, and vice versa. This is done in the paper referred to. Moreover, with the suppositions just made as to the numbers a, §' being odd and a, § even, it is obvious that TO' is odd or even, according as TO is, and n' according as n is, which shows that we can likewise satisfy TO

+ \ to + n + % v + 2r6 =TO'+ \ a + n' +1 v ;

and thus the denominator of fax is also reducible to the required form. Now proceeding to the immediate object of this paper, a, §, a, §', and consequently to', v are to a certain extent indeterminate. Let A, B, A', B' be a particular set of values of a, §, a, §', and 0, P the corresponding values of to', v'. We have evidently A, E odd and A', B even. Als,o AB'-A'B= p, fxB' — vA' — L'p, fiB — vA = Lp, 0=-(Aa lT=-(A'to P

+ Bv),

134

ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.

[23

By eliminating a, v from these equations and the former system, it is easy to obtain

v'^a'O + b'U, where

a = ~(<xB' - § A'), px ' a' = - (a'B' - S'A'),

b= py

--(aB-SA), '

V = - - (a'B - €'A).

The coefficients a, b, a', V are integers, as is obvious from the equation fi(a'B—§'A) = p(L'a — lA'), and the others analogous to i t ; moreover, a, b' are odd and a', b are even, and ab' - a'b = \ (A'B - A'B) (a§' - a'g); that is

ab' — a'b = l.

Hence the theorem,—" The general values a>', v of the complete functions are linearly connected with the particular system of values 0, U by the equations, tPc is precisely the same, whether 0, U or G/, V' be taken for the complete functions. In fact, stating the proposition relatively to <£>%, we have,—" The inverse function x is not altered by the change of as, v into a/, v', where to' = aw + §v, v = a'co + §'v, and a, §, a', §' satisfy the conditions that a, §' are odd, «', § even, and aS' — a!S = 1." This is immediately shown by writing Bsi + nv = m'a> + n'v', or

m = m'a + n'a', n — m'€ + n'§'.

It is obvious that to each set of values of m, n there is a unique set of values of m', n', and vice versa: also that odd or even values of m, m' or n, n' always correspond to each. It is, in fact, the preceding reasoning applied to the case of p=l. Hence finally the theorem,—" The only conditions for determining to', v are the equations a>' = - (aw + Sv),

v'= -(a'w + §'v),

where a, §' are odd and a', § even, and aS' - a'§ = p,

fig' - va! = Vp,

fiS-va

= Ip,

I and V arbitrary integers: and it is absolutely indifferent what system of values is adopted for «', v, the value of fax is precisely the same."

23]

ON THE TRANSPOBMATION OF ELLIPTIC FUNCTIONS.

135

We derive from the above the somewhat singular conclusion, that the complete functions are not absolutely determinate functions of the modulus; notwithstanding that they are given by the apparently determinate conditions, ,-

dx

1" = dx

*.-/;

In fact definite integrals are in many cases really indeterminate, and acquire different values according as we consider the variable to pass through real values, or through imaginary ones. Where the limits are real, it is tacitly supposed that the variable passes through a succession of real values, and thus to, v may be considered as completely determined by these equations, but only in consequence of this tacit supposition. If c and e are imaginary, there is absolutely no system of values to be selected for w, v in preference to any other system. The only remaining difficulty is to show from the integral itself, independently of the theory of elliptic functions, that such integrals contain an indeterminateness of two arbitrary integers; and this difficulty is equally great in the simplest cases. Why, d priori, do the functions dx

,

/" dx

log x= j -

contain a single indeterminate integer ? OBS.

I am of course aware, that in treating of the properties of such products as x \ 1H , it is absolutely necessary to r pay attention to the relations between J J J mm + nvj the infinite limiting values of m and n; and that this introduces certain exponential factors, to which no allusion has been made. But these factors always disappear from the quotient of two such products, and to have made mention of them would only have been embarrassing the demonstration without necessity.

(

136

[24

24. ON THE INVEKSE ELLIPTIC FUNCTIONS. [From the Cambridge Mathematical Journal, t. iv. (1845), pp. 257—277.] THE properties of the inverse elliptic functions have been the object of the researches of the two illustrious analysts, Abel and Jacobi. Among their most remarkable ones may be reckoned the formulae given by Abel (QSuvres, t. i. p. 212 [Ed. 2, p. 343]), in which the functions a, fa, Fa, (corresponding to Jacobi's sin am. a, cos am. a, Aam. a, though not precisely equivalent to these, Abel's radical being [(1 — cV) (1 + eV)]i, and Jacobi's, like that of Legendre's [(I — *2) (1 — k2a?)]i), are expressed in the form of fractions, having a common denominator; and this, together with the three numerators, resolved into a doubly infinite series of factors; i.e. the general factor contains two independent integers. These formulae may conveniently be referred to as "Abel's double factorial expressions" for the functions <£, /, F. By dividing each of these products into an infinite number of partial products, and expressing these by means of circular or exponential functions, Abel has obtained (pp. 216—218) two other systems of formulae for the same quantities, which may be referred to as "Abel's first and second single factorial systems." The theory of the functions forming the above numerators and denominator, is mentioned by Abel in a letter to Legendre (CEuvres, t. n. p. 259 [Ed. 2, p. 272]), as a subject to which his attention had been directed, but none of his researches upon them have ever been published. Abel's double factorial expressions have nowhere anything analogous to them in Jacobi's Fund. Nova; but the system of formulae analogous to the first single factorial system is given by Jacobi (p. 86), and the second system is implicitly contained in some of the subsequent formulae. The functions forming the numerator and denominator of sin am. u, Jacobi represents, omitting a constant factor, by H (u), © (it); and proceeds to investigate the properties of these new functions. This he principally effects by means of a very remarkable equation of the form Z© (u) - % A u? + BJ0 du. /„ du sin 2 am u,

24]

ON THE INVERSE ELLIPTIC FUNCTIONS.

137

{Fund. Nova, pp. 145, 133), by which © (u) is made to depend on the known function sin am. u. The other two numerators are easily expressed by means of the two functions H, ©. From the omission of Abel's double factorial expressions, which are the only ones which display clearly the real nature of the functions in the numerators and denominators ; and besides, from the different form of Jacobi's radical, which complicates the transformation from an impossible to a possible argument, it is difficult to trace the connection between Jacobi's formulae; and in particular to account for the appearance of an exponential factor which runs through them. It would seem therefore natural to make the whole theory depend upon the definitions of the new transcendental functions to which Abel's double factorial expressions lead one, even if these definitions were not of such a nature, that one only wonders they should never have been assumed a priori from the analogy of the circular functions sin, cos, and quite independently of the theory of elliptic integrals. This is accordingly what I have done in the present paper, in which therefore I assume no single property of elliptic functions, but demonstrate them all, from my fundamental equations. For the sake however of comparison, I retain entirely the notation of Abel. Several of the formulae that will be obtained are new. The infinite product

•n (!+-£•)

(i),

mco

where m receives the integer values + 1 , +2,... ±r, converges, as is well known, as TTOC

r becomes indefinitely great to a determinate function sin — of x; the theory of which might, if necessary, be investigated from this property assumed as a definition. thus naturally led to investigate the properties of the new transcendant it = « n i l lH ; \ nidi + nvij

We are

(2) :

m and n are integer numbers, positive or negative; and it is supposed that whatever positive value is attributed to either of these, the corresponding negative one is also given to it. i = A/ (— 1), a> and v are real positive quantities. (At least this is the standard case, and the only one we shall explicitly consider. Many of the formulas obtained are true, with slight modifications, whatever to and v represent, provided only to : vi be not a real quantity; for if it were so, mto + nvi for some values of m, n would vanish, or at least become indefinitely small, and u would cease to be a determinate function of x.)1 Now the value of the above expression, or, as for the sake of shortness it may be written, of the function r

\

(m, n)

(3),

1 I have examined the case of impossible values of w and v in a paper which I am preparing for Crelle's Journal. [The paper here referred to is [25], actually published in Liouville's Journal].

C

18

138

ON THE INVEBSE ELLIPTIC FUNCTIONS.

[24

depends in a remarkable manner on the mode in which the superior limits ofTO,n are assigned. Imagine m, n to have any positive or negative integer values satisfying the equation (m\ n*)
(4).

Consider, for greater distinctness,TO,n as the coordinates of a point; the equation cf> (TO2, v?) = T belongs to a certain curve symmetrical with respect to the two axes. I suppose besides that this is a -continuous curve without multiple points, and such that the minimum value of a radius vector through the origin continually increases as T increases, and becomes infinite with T. The curve may be analytically discontinuous, this is of no importance. The condition with respect to the limits is then that TO and n must be integer values denoting the coordinates of a point within the above curve, the whole system of such integer values being successively taken for these quantities. Suppose, next, u' denotes the same function as u, except that the limiting condition is '(m\ n*)
(5).

The curve (/>' (TO3, w2) = T' is supposed to possess the same properties with the other limiting curve, and, for greater distinctness, to lie entirely outside of i t ; but this last condition is nonessential. These conditions being satisfied, the ratio v! : u is very easily determined in the limiting case of T and T' infinite. In fact

=nnji + r ^ l u

or

{

(TO,

(6),

n)\

J« = 2 s J i + _!L_l u

{

(7),

(m, n))

the limiting conditions being (m% n*)>T

Now

j ji+ _ ^ _ l ( (m, n))

=

(8),

* $. * ^ + (m, n) (m, nf

(9),

= x.XZr^-%a?.t$Y-i^ + (10), u (m, n) i (TO, tif or, the alternate terms vanishing on account of the positive and negative values destroying each other, = J a . 2 2 -i-^-Jfl-.SST-^;-

u

z

(TO,

nf

4

(m, n)4

(11).

In general , M)=//V|T(TO,

n)dmdn + P

(12),

24]

ON THE INVERSE ELLIPTIC FUNCTIONS.

139

P denoting a series the first term of which is of the form Cyjr (m, n), and the remaining ones depending on the differential coefficients of this quantity with respect to m and n. The limits between which the two sides are to be taken, are identical. In the present case, supposing T and T indefinitely great, it is easy to see that u the first first term of the expression for I — is the only one which is not indefinitely t small; and we have

where .

=

ff dmdn

[[

dmdn

., ..

^14);

^ JJK^rJJ(mM+w^ the limits of the integration being given by T f (m2, O < 2". Some particular cases are important.

(15),

Suppose the limits of u' are given by

and those of u, by mW + « V < 2 *

(17);

we have

{

L

aJ

[T+nvi

2 f

f

V {T* ~ nV) + nvi

-*J(P-

nV) + nvi

-T+ nvi

T

or, in this case, w' = «

(19).

Again, let the limits of u' be m2co2
and those of u,

mW<E 2 , dmdn (mco+nvif If, 0) J

f 1 \R' + nvi

«V
(20),

«V<,Sf2

(21).

1 R + nvi

1 [

- R

1 L

+ nvi

— R' + nvi) '

18—2

140

ON THE INVERSE ELLIPTIC FUNCTIONS.

[24

where the limits are mV < S'2, for the terms containing R', reV < S2, for the terms containing R, ~

2 R'+S'i R-Si avi R'-S'i R + Si

{

(x\),

^ ,

(OV

X

}

'

n | , ±1

M

the arcs A, \ ' being included between the limits 0, \TT. Hence u'=ue*{k'-x)™ j.- i .e 8' 8 , In particular it ^ = ^ 5 , u — u. MR T

TJ?

If

(24).

8' A S .. , ,_ 2 .„ S' S , ™ = 0, ^ = 1 , M = ue~itix ; 11 -g> = 00, -== 1, It It MM

u' = ue^^: where 8 — — , for which quantity it will continue to be used. cov We may now completely define the functions whose properties are to be investigated. Writing, for shortness, (m, n) = mm + nvi

(A),

{m, n) = (m + if) a> + nvi, (m, n) = mm + (n + %) vi, (m, n)=(m + fj eo + (n + \) vi ; we may put

yx=xuuh + (jn, n) gx = n n | i +(m,x n) Gx= n n l i + (TO,x n) &X=

IUIH^TT:,

(

(TO,

n)

the limits being given respectively by the equations mod. (TO, n) < T, mod. (m, n) < T,

mod. (TO, n) < T, mod. (m, n) < T,

T being finally infinite. The system of values m = 0,ra= 0, is of course omitted in yx. The functions yx, gx, Gx, (Six, are all of them real finite functions of x, possessing properties analogous to that of u. Thus, representing any one of them by Jx, we have (C),

24]

ON THE INVERSE ELLIPTIC FUNCTIONS.

141

where J±pX is the same as Jx, only for J^x the limits are given by mW or (m + | ) 2 to2 < R\ mV or (n + ^) 2 u 2 <£, (i?, 8, and -^ infinite), and for J-Px, by the same 8 formulas, (R, 8, and -5 infinite).

It is to this equation that the most characteristic

properties of the functions Jx are due. The following equations are deduced immediately from the above definitions: 7(-a;)

=-7a;,

7 (0)=0,

g{-x)=gx,

g(0) = l,

O(-x) = Gx, CE (-x) = (Six

(D),

0(0) = 1, ffi(O) = l,

Suppose and v,—then changing x into xi, and interchanging m and n, by which means the limiting equations are the same in the two cases, we obtain the following system of equations: gx (xi) = Ox, Or (xi) = gx, <&l(xi) = (&x; or otherwise, (F),

7 (xi) = vfix g (xi) = 01x, 0 (xi) = g-,x, CEr {xi) = ffixar,

equations which are useful in transforming almost any other property of the functions J. The functions J^x are changed one into another, except as regards a constant multiplier, by the change of x into x + -x. This will be shown in a Note, or it may be seen from some formulae deduced immediately from the definitions of the functions Jpx, which will be given in the sequel1. Observing the relation between Jx and we have in particular (

C

^

(

0

)

,

( 1

Not given in the present paper. been the formulae (M) p. 144.]

[The Note was given, see p. 154, and the formula referred to must have

142

ON THE INVERSE ELLIPTIC FUNCTIONS.

[24

where A, B, G, D, are most simply determined by writing « = 0, x = — -^. Putting at the same time epw2 = e ~ = qr1,

B—sr'-

whence also

Similarly, the functions J-$% are changed one into the other by the change of x into x + \ id. We have in the same way

ffi (a? +

Y)

=

Whence

where e ^ ^ e " =q~1. It is obvious that the relation between q and ql is Iq.lq1 We obtain from the above

24]

ON THE INVERSE ELLIPTIC FUNCTIONS. Also, by making x = -^ in the expression for y(x+~) -y),

and # = ~

143 m tnat

f° r

we have

and the same or an equivalent one would have been obtained from the functions g, G, €&. By combining the above systems, we deduce one of the form

0V,

and, observing the equation e'3""'* = e" = (— 1), with the following values for the coefficients,

ir—<-i).. B -.. T (2) +7 g),

Collecting t h e formulae w h i c h connect 7 ( ^ ) , 7 ( ^ - ) ,

!)«(!)='•"'•

these a r e

144

ON THE INVERSE ELLIPTIC FUNCTIONS.

[24

And by the assistance of these B"C" + A"D"=

B'D' + A'C

=CD + AB = -1

(28),

A' B' + C'D' = - A"B" + G"D" = - 7 3 g ) - ffi2 g ) , 4"C" -=- F'D" = -A'C

+B'D' = vi\ 2J'

which will be required presently. It is now easy to proceed to the general systems of formulae,

7 {x + (m, n)) = (— l) m +». ©ryaj j (m, w)} = ( - l ) ™

(S[{x + (m, n)} = (J) = < _ ! ) « ( » » + i) gfaUm + Doi—nvi] „ — Jm 2 —Jm^

7 {x + (m, «)} = ( - l ) m + m . <&Agx ,

g{x + (m, n)} = (-l)m G{x+ (m, »)} = ( - 1)"

. .

<&{x + (m, w)} = \£r

=

/ _ ! ) » » ( n + J ) e/3a; [m
7 {» + (m, «)} = ( - l) m + m . g{x + (m, n)} = ( - l ) m G {* + (m, n)} = ( - 1 ) " ffi {x + (m, n)} =

7 {«+ (m, n)} = ( - l ) m + g {x + (m, «)} = ( - l ) m (^ {x + (m, n)} = ( - l)n

. ilC'gx ,

ffi {x + (m, n)} =

. D,D"yx.

24]

ON THE INVERSE ELLIPTIC FUNCTIONS.

145

Suppose x = 0, we have the new systems,

®0 = (-l)mnq{-im'q-in2

(Mbis).

7 (ro, n) = 0, g{m,

y (m, ») = ( - 1 )

m+

" ©0,

m

n) =

(-l) ®0,

G(m, n) = ( - l ) » © 0 , ©

(TO,

») =

0O.

7 (TO, n) = (— l) m +» $ 0 4 , gr (m, n) = 0,

(TO,

^ (m, n) = ( - l)

n) =

7 (m, n) = ( - i jr (m, n) = ( - l ) m

^ 0 B',
G (m, n) = 0, ©

(TO,

n) =

7 (TO,

n)=(-l)m+na0A",

g(m,n) = (- l)m

&$",

6(m, n) = (-l)»

n0C",

© (m, n) = 0,

05' (TO, n) == fi0D".

We obtain immediately, by taking the logarithmic differentials of the functions yx, gx, Gx, (Six, the equations y'x-i-
(TO, M)}"1, TO

= 0,

n = 0 admissible,

(iV),

1

(TO, n)}- ,

(&'x + <&x = 2 2 {x - (TO, n)}-1, the limits being the same as in the case of the factorial expressions. Consider an equation gx Gx + yx <&x = 2 2 [ 8 {x - (TO, n)}-1 + 3&{x- (TO, n)}"1] C.

(29), 19

146

ON THE INVERSE ELLIPTIC FUNCTIONS.

we have

@l=g(m, n) G (m, n) -i-y' (m, n)(& {m, n) = 1

[24 (30),

33 = g (m, n) G (m, n) -s- 7 (m, n) <&' (m, n) = B"G" 4- A"D" = - 1 ... (31). (The application of the ordinary method of decomposition into partial fractions, which is in general exceedingly precarious when applied to transcendental functions, is justified here by a theorem of Cauchy's, which will presently be quoted.) We have thus gxQx -r- yxt&x = (y'x -r- yx) — {I&'x -=- (fox), and similarly -T- yX Gx = (yX-r-

JX) — (G'X -r Gx),

(0),

Gx($ix -=- yx gx = ( y'x •+• yx) — ( g'x -r gx), - &yx($ix -r-gx Gx = ( g'x -r gx) — (G'x -4- Ga;), e27a; £f« +Gx&x

= (G'x + Gx) - (ffi'a;-=- CEr«),


6" Eliminating the derived coefficients, (33),

Adding these equations, 62 = e2 + c2, or 6 = V(e2 + c2), in which sense it will continue to be used. Also,

g'x = <&?x - cyx,

(P).

G*x =i&*x + e>y*x. Suppose

x = yx ~- (fax,

,

(Q),

fx =gx -nCBra;, Fx =Gx+(&x, then

f2x=l-d>%

(R),

24]

ON THE INVERSE ELLIPTIC FUNCTIONS.

and also

$'x = fx Fx ,

147 ($),

F'x = etyxfx Hence, putting for fx, Fx, their values, J(1 - C24>2x) (I + e*cfr>x)

or writing $# = y, and integrating,

d x=l

v

{U)i

s/(i-cy)(i+ey) or which shows that $ is an inverse elliptic function. The equations which are the foundation of the theory of the functions <£, f F, are deduced immediately from the equations (S). (Abel, (Euvres, torn. i. p. 143 [Ed. 2, p. 268.]) These are

so that from this point we may take for granted any properties of these functions. We see, for instance, immediately, vi

;SJ'

whence vi

2

[e

^

dy

(if),

_

v

— '

-

ios/(l-cy)(l+ey)'

OT1 -

2

=

which give the values of w, v in terms of c, e; values which may be developed in a variety of ways, in infinite series. We may also express 7 ( „ ) , &c, and consequently A, B...&C, by means of the quantities c, e. We have only to combine the equations vi

i

„ fa>\

^» fa\

b

fvi\ ^ fvi\ _ b 19—2

148

ON THE INVERSE ELLIPTIC FUNCTIONS.

[24

with the former relations between these quantities, and we have

tf-*«-*r*> (F),

I) = 6-* c* gr»

® (I) = 6~* e* 2

G" = (-1)1 which are to be substituted in any formula into which these quantities enter. The following is Cauchy's Theorem, (Exercises de Math. t. n. p. 289). " If in attributing to the modolus r of the variable z = r {cos^ + V(— l)sinp}

(35),

infinitely great values, these can be chosen so that the two functions 2

'

Tz

(36)

'

sensibly vanish, whatever be the value of p, or vanish in general, though ceasing to do so and obtaining finite values for certain particular. values of p; then

the integral residue being reduced to its principal value." To understand this, it is only necessary to remark that question is the series of fractions that would be obtained by decomposition; and by the principal value is meant, that all taken, the modulus of which is not greater than a certain afterwards made infinite.

the integral residue in the ordinary process of those roots are to be limit, this limit being

Suppose now fa is a fraction, the numerator and denominator of which are monomials of the form (7a;)' (goc)m..., I, m... being positive integers, and of course no common factor being left in the numerator and denominator. Let X be the excess of the degree of the denominator over that of the numerator. Suppose the modulus r of (z) has any value not the same with any of the moduli of (m, n), (m, n), (m, n), (m, n)

(38).

24 J

ON THE INVERSE ELLIPTIC FUNCTIONS.

149

Then we have r (cosp + isinp) = meo +nvi+ 0

(39),

6 being a finite quantity, such that none of the functions JO vanish, m and n are the greatest integer values which allow the possible part of 0 and the coefficient of its impossible part to remain positive. We have therefore mW +raV= / * - . ¥

(40),

M being finite; or when r is infinite, at least one of the values m, n is infinite. The function fz reduces itself to the form emA+n

q j

where F is finite. Hence q1 and q being always less than unity, fz, and consequently both \ {fz+f(— z)) and ^- {fz— f(— z)} vanish for r=oo, as long as \ is positive. In the case of A, = 0, the conditions are still satisfied, if we suppose fx to denote an uneven function of x: for when X = 0, the index of exponential in the above expression vanishes, or fz is constantly finite. But fz being an odd function of z, fz+f(—z)

= 0. And ^- {fz— / ( — z)) vanishes for z infinite, on account of the z in

the denominator: hence the expansion is admissible in this case. But it is certainly so also, in a great many cases at least, where fz is an- even function of z\ for these may be deduced from the others by a simple change in the value of the variable. For instance, from the expansion of
(mw+nur)' q^m* qfrri*

(42) ;

or, if a = h + M, this becomes, omitting a finite factor, 6

- £ m 2 (A/5—A) w a -

which vanishes if h? + & < X2/32, i.e. the modulus of a is less than \fi. case is admissible when the series is convergent.

The limiting

We obtain in this way a very great variety of formulas. For instance, ». n)Hb{m,n) j ^ ^ n l {x — (m, m)}"1] (m, n)*+» (m, n) Jm-n

6£a

(m, n)2+b (m, n) ,n)

(A'),

150

ON THE INVERSE ELLIPTIC FUNCTIONS.

[24

in which the modulus of a must not exceed yS: in the limiting cases, for a — /3, b must be entirely impossible, and for a = — /S, b must be entirely real. The formulae for yx are 8 e" <m' "> {* - (m, w)}-1 (44), -=- 7a; = 2 2 ( - I)" 1 "* ft"1' e6 <m- »' {x - (m, n)}'1; and for 6 = 0, 1

' { « - ( m , TI)}-

w!

1

(45), 1

{« - (m, m)}- .

Next the system, ffi«-r-7a; =tt(-l)m+n{x-(m,

TO)}-1

5r« -r yx ••= 2 S (— l ) m

{« — (m, n)}"1,

Gx + yx = 2 2 ( - l) w yx+gx = - &

{« - (m, n)}-1;

Gx + gx = - c- 1 2 2 ( - l) m + m [x - ( i , <&x + gx = -ft" 1 2 2 ( - 1)™

n)}'1,

{# - (m, n)}" 1 ;

-T- (?» = - b-1 e'122 (-.1)™ {* - (m, n)}- 1 ,

7«

gx+Gx= ffias -5- ©« = 7a;

ier1 2 2 ( - l ) m + B {x- (m, n)}- 1 , 16"1 2 2 ( - 1)"

^(Six^ic-^er122

{x - (m, n)}"1;

( - l)m+» {» - (m, n)}~1,

gx +<§ix= e-1 2 2 ( - 1)»

{x - (m, n)}"1,

fe -=-©*= ic- 1 2 2 ( - l) m

{x - (m, n)}-1,

which is partially given by Abel. We may obtain, in like manner, expressions for the functions

t yxgx

(six

termS of this

(twelve)

IB'),

(four)

(GO,

24]

ON THE INVERSE ELLIPTIC FUNCTIONS.

151

each of them, except (E'), (the system for which, admitting no exponential, has already been given,) multiplied by an exponential ei^2+^y the limits of a being + 2/3, + /3, ±0, ± 3/3, + 2/3, + 4/3. For the limiting values, b must be entirely impossible for the superior limit, and entirely possible for the inferior one. Thus the last case is

yxgxGx(8ix 2 2 2 "

= 2 2 [e*« (»»• »>*+» <»•»> ^ - 2 2 [ e 4 a {™" n)1+b <"»•"» ^

(OT+ )S

* g2"2

+ 2 2 [e* o (m ,»)•+» (m, n) qt nfl

{« - O > w)}" 1 ] {« - (m, w)}- 1 ]

^ («+i, • ^ _ ( m > jj)j-i]

- 2 2 [e*a (™- »)2+6 ^ 2 c»+i)a g2 (»+J)a {# - (m, n)}" 1 ]; in particular

2 '

k - ( m , n)}-1

+ 2 2 s^ 8

{* - (m, n)}"1

1

(46),

+ 2 2 g1<2m+1)2 {«; - . (TO, n)}-1, or the analogous formula obtained by changing 0, qly m into — /3, j , n. The function (f>% which is even, and for which \ = 0, cannot be expanded entirely in a series of partial fractions: but (x — a)" 1 2x may be so expanded. Multiply by (x — a), the second side has for its general term (x - a) (Mx + N){x- (mx n)}-% equivalent to K'+{M'x + N'){x-(m, n)}-\ Summing all the K"s, we have an equation of the form fix = A + 2 2 [L [x - (m, n)}-* + M{x - (m, n)}-1]

(47).

To determine the coefficients as simply as possible, change x into x + \w + fyuoi, - e-2 c~2 (0*)- 2 = A + 2 2 [L {x - (m, TO)}"2 +M{XL=-e-ic~*[{x-(m,

(m,

ri)}-1]

n)}1 (^«)~2],x = (m, n)

(48), (49),

M = - e-* c~a dx [{as - (m, n)}2 (t£«)-2], or writing x + (m, n) for x, and therefore x = 0 in the values of L and M, L = - e-2 c~2 {*2 (x)-2} = e" 2 cr 2 2

2

2

(50),

2

i f = - e~ c- 8s {a (
<^x= A -e~2c~222

{ « - (m, n)}" 2

(51).

152

ON THE INVERSE ELLIPTIC FUNCTIONS.

[24

Integrating this last equation twice, J(,dxJodx^is = ^Axi + e-!ic-^'Z%l{ic-(m,n)}

(52),

(Srx = e-ie^Ax'+e^fodxJodx^x

(53),

or

an equation from which it is easy to determine the coefficient A. Suppose for a moment <j>/x=J0//x = jo,xdx; then, since $ 2 {x + co) — <$>x = 0, 4>, {x + w)—$x = #,©,

<£„ (x + co) - $tx = <£„« + xfaco.

But similarly — x) = 0 ; whence

4>F + ^ (ffl - x) = ^ w > //* - ^>// ( « - « ) + „« = xfaw ; whence, writing x — g-,

£

&(!)

or ^

Hence

© ( « + <») = e " * ^ ^-*w*' »w)i (2^+I»!»

But

C5(a; + a)) = e^ir^1-i€fra; = 6W(^+'»2) © «

ffix

(54). (55);

or, comparing these,

{ I ® } * ^ ( | )

(56)

'

(57),

or writing M =—

Mxdx

(58),

CO J o then

( 3 T # = e
(•/'•) .

which is the formula corresponding to the one of Jacobi's referred to at the beginning of this paper. Analogous formulae may be deduced from it by writing x + ~ , or x + —, or x + •£• + -g-, instead ot x.

The following formulae, making the necessary changes of notation, are taken from Jacobi. We have ,,/

N

4><j)a fa Fa d>x fxFx

_^ ( - ,- 0 )»^Z__tZ_

,__,

(59),

24]

ON THE INVEESE ELLIPTIC FUNCTIONS.

whence

J0{^(x

+

a ) - <j>>(x - a)} dx = f

^

^

153 ^

(60),

the first side of which is J_a2(x- a)dx-2J0^ada

(61).

Hence, multiplying by e2c2, and observing the value of ffia;,

&' (x + a) _ <&'{x-a) _ &a_ _ 2e*c* fa Fa fa^x € f ( * + a)

CBf(*-a)

©a~

1 + e2c2<£2a02c

*• '"

If in this case we interchange x, a and add, <&'x &a -?^- + -^r Sir* Cera

(&'(x + a) t Tjty—;—v = ^ * « # ^ ( i + « V5(x + a) -r T- T \

.... (63). \ /

[By subtracting, we should have obtained an equation only differing from the above in the sign of a.] Integrating the last equation but one, with respect to a, l<& {x + a) + l(St (x-a)the integral being taken from a = 0.

2Mx - 2Ma =1(1+

eWftxfia),

Hence

€Sr (x + a) $f (x - a) = eJ2«ffi2a (1 + e^xpa) or

(64);

C

whence also

7 (x + a) y(x-a) = g(x + a) g(x-a) = g'x&a - c2g2a G(x+a) G(x-a)=G*x(8,*a + e2G2a<§t2x,

(y )>

these equations being obtained from the first by the change of x into x + s ' , x + -^, x

+ a + ~n • 1 n e y form a most important group of formulae in the present theory. By

integrating the same formulse with respect to x, and representing by n (x, a) the , f — e2c2 Jacobi obtains 5 Jo 1 + e2c2<£2a2«

an equation which conducts him almost immediately to the formulse for the addition of the argument or of the parameter in the function LT. This, however, is not very C. 20

154

ON THE INVERSE ELLIPTIC FUNCTIONS.

closely connected with the present subject.

[24

For some formulas also deduced from (63),

{**/ \ ,«•, { is expressed in terms of the function d>, see by which -^sA J r r <St (x + a) (S (y + a) (ffif (x + y — a) Jacobi. NOTE.—We have x yaX = xUu(l+ (m, n)J'

- nn[i+ (m,

n))'

the limits of n being ±q, and those of m being ±p, in the first case, and p, —p — 1, fry

in the second case.

Also - = 00 . ? We deduce immediately

i + V4r-nnfi + r ^ ) + ; [ (TO, n) J V (TO, « ) / 2 (paying attention to the omission of (TO = 0, ra = O) in 7,3a;, and supposing that this value enters into the numerator of the expression just obtained, but not into its denominator). This is of the form

but the limits are not the same in this product and in g$x. In the latter m assumes the value —p — 1, which it does not in the former; hence

and the above product reduces itself to unity in consequence of all the values assumed by n being indefinitely small compared with the quantity (p + \); we have therefore 7/3

and similar expressions for the remaining functions. To illustrate this further, suppose we had been considering, instead of y^x, the function
) ( p /

+

The divisor of the second side $

( V

nvi

24]

ON THE INVERSE ELLIPTIC FUNCTIONS.

where the extreme values of n are infinite as compared with p.

155

This may be reduced to

- sin —. {x - (p + £) co] -¥• sin (p + £) w,

H-6"

=6

",

neglecting the exponentials whose indices are infinitely great and negative. the value of /3 this becomes e~^x, and we have

Observing

7-* (<*+§) = «"""• ^ - * - * : a result of the form of that which would be deduced from the equations y_p x = e&yp x, 'Y^(a) + -) = Agpx.

I t is scarcely necessary to remark that y-pw has the

same relations to the change of % into x + -x as yp x has to that of x into x + -^.

20—2

156

[21

25. MEMOIRE SUE LES FONCTIONS DOUBLEMENT PEEIODIQUES. [From the Journal des Mathematiques (Liouville), torn. x. (1845), pp. 385—420.] des plus beaux re"sultats des recherches de l'illustre Abel, dans la the"orie des fonctions elliptiques, consiste dans les expressions qu'il obtint pour les fonctions inverses a, fa, Fa (e'quivalentes a-peu-pres a sin ama, cos ama, Aama) en forme de fractions avec un de'nominateur commun: ce d^nominateur et les trois numeVateurs etant chacun le produit d'une suite infinie double de facteurs. On ne sait pas a quel point Abel avait' pousse" 1'investigation des proprie'tes de ces nouvelles fonctions; on trouve seulement, dans une Lettre a Legendre, imprimee parmi ses CEuvres [t. n. p. 259, Ed. 2, p. 274], qu'il s'en e"tait occupe". Depuis, les fonctions H, ©, qui sont essentiellement les memes que ces fonctions d'Abel, ont e'te' l'objet des savantes recherches de M. Jacobi, a qui Ton doit, en particulier, la belle formule log © (a) - log © (0) = la 2 (l - -=/) - &21 da \ da sin2 ama, V ti-1 Jo Jo qui est vraiment fondamentale, et sur laquelle on peut dire que sa thdorie est basee. Mais les expressions qu'obtient M. Jacobi pour les fonctions H, ©, sont sous la forme d'un produit d'une suite infinie simple de facteurs, ce qui ne met pas a. beaucoup pres si bien en evidence la vraie nature de ces fonctions que les expressions d'Abel; celles-ci sont, en outre, si analogues aux formules en produits infinis des fonctions circulaires, que Ton est seulement e'tonne que personne ne se soit avise jusqu'ici de les poser, a priori, comme les definitions les plus simples des fonctions doublement pe"riodiques, pour en deduire la the'orie de ces fonctions. C'est de cette maniere que je me propose de traiter ici la question. Je prends pour definitions les formules d'Abel, en supposant, pour plus de ge"neralite, que les fonctions completes 12, T (K, K' de M. Jacobi) sont chacune de la forme A + B J — 1 (ce qui donne lieu a, quelques integrations assez delicates). Et de ces seules Equations, sans me servir en rien de la

25]

MEMOIRE SUE, LES FONCTIONS DOUBLEMENT PERIODIQUES.

157

the"orie des fonctions elliptiques, je deduis les proprie'tes fondamentales des fonctions en question, et de la, des fonctions elliptiques. On a ainsi quatre fonctions a considerer au lieu des deux H, ©, dont l'une est pour ainsi dire analogue a. un sinus et les autres a des cosinus. Mais ce qu'il y a de remarquable, c'est l'apparition d'un facteur exponentiel qui entre presque partout. On le preVoyait d'apres les formules de M. Jacobi, mais ces formules n'expliquent pas, ce me semble, pourquoi ce facteur s'y rencontre: mon analyse le fait voir de la maniere la plus satisfaisante. Ce facteur resulte, en effet, de ce que, pour les produits infinis doubles, il ne suffit pas, pour obtenir un rdsultat determine, d'attribuer aux deux entiers variables des valeurs quelconques depuis — oo jusqu'a oo, me1 me en supposant l'e'galite' des valeurs positives et negatives; il faut, en outre, etablir une relation entre les valeurs infinies que recoivent les deux variables. Mais dans les produits que je considere, on demontre qu'en supposant toujours cette e'galite' des valeurs positives et negatives, quelque liaison que Ton etablisse entre les valeurs infinies, il rdsulte toujours la meme valeur du produit, a un facteur exponentiel pres, dont l'indice est le carrd de x, multiplie par une constante dont la valeur s'exprime au moyen d'une integrale definie double, et qui de'pend de la liaison etablie entre les valeurs infinies des variables. C'est-a-dire qu'en multipliant par un facteur exponentiel de cette forme, convenablement choisi, on peut changer a volonte la relation en question sans affecter la valeur du produit. Voila l'idee fondamentale du Me'moire qui suit. En me servant de quelques formules de M. Cauchy, relatives a. la decomposition des fonctions en fractions simples, j'etablis d'une maniere rigoureuse des relations entre les trois quotients de mes quatre fonctions, qui sont les memes par lesquelles Abel demontre les proprie'te's fondamentales des fonctions $, f, F. Ces formules une fois obtenues, on peut supposer connue toute la the"orie des fonctions elliptiques. Ces theoremes de M. Cauchy me conduisent, en outre, a un grand nombre de nouvelles formules qui contiennent des suites infinies doubles. Parmi celles-ci, il y en a une qui me fournit la demonstration du theoreme deja cite de M. Jacobi, the'oreme duquel il de"duit une foule de resultats interessants. Je finis en citant ceux qui se rapportent de plus pres aux fonctions dont je parle. J'espere reprendre une autre fois la consideration d'une autre partie de la thdorie, dans laquelle j'entrevois des conclusions inte'ressantes. Soient il, T des quantity's finies quelconques, assujetties a la seule condition que la fraction il : T ne soit pas reelle. En representant par m, n des entiers positifs ou negatifs quelconques, mettons, pour abreger, mil +nT = (m, n)

(1),

et considerons une expression de cette forme

u^auil

+ j-^—X I (m, «)J

(2),

ou le symbole II denote, comme a l'ordinaire, le produit d'un nombre infini de facteurs que Ton obtient en donnant a m, n des valeurs entieres quelconques, depuis — oo jusqu'a

158

MEMOIRE SUE LES FONCTIONS DOUBLEMENT PEEIODIQUES.

[25

+ oo , en excluant seulement la combinaison (m = 0, n = 0). Pour qu'une telle expression soit finie, il faut que pour chaque combinaison de valeurs de m, n, il y en ait une autre des mSmes valeurs avec les signes contraires. Cependant, comme on l'a deja explique, cela ne suffit pas pour rendre de'termine'e la valeur de u. Soit (/> une fonction de m, n qui ne change pas en changeant a la fois les signes de ces deux quantite's; et imaginons que l'equation

represente une courbe ferme'e, dont tous les points s'e'loignent, au cas limite de T = oo, d'une distance infinie de l'origine. Cela pose', en donnant a m, n des valeurs entieres qui satisfassent a cette condition < T, et puis faisant T = oo, on obtient pour u une valeur parfaitement de'termine'e, qui depend de la forme de la fonction . Soit v! ce que devient la fonction u en changeant seulement liquation aux limites dans l'equation analogue

on peut, pour simplifier, supposer que la courbe repre'sente'e par cette equation soit situde entierement en dehors de celle que repre'sente l'e'quation

mais cela n'est pas essentiel. II est facile de trouver une relation tres-simple qui existe entre ces deux expressions u' et u. En effefc,

«' : « = n | l +- °° (m,

en donnant a m, n des valeurs qui satisfassent a la fois aux deux conditions > T, <j>' < T. Done, en conside'rant toujours ces valeurs, logtt'- log« = logn | l + 7-^-vl S log 7-^-vl ==S log (l + 7 \ = ^T—S - ^ S T - ^ + ... (m, «)j [ (m, n)) ** (m, n) 2 ^ (m, rif 5 8 { (m «)j Dans cette expression, les termes qui contiennent les puissances impaires x s'eVanouissent, a cause des valeurs e'gales positives et negatives. Mais puisque, a, la limite, m, n ne re§oivent que des valeurs infinies, on peut negliger les termes multiplies par cfi, &c. Done log u' - log u = - \a? 2

T

r2,

ou enfin, logu'-\ogu = - %Ax\ u' = ue-iAxi

(3),

ou j'ai repre"sentd par e la base du systeme hyperbolique de logarithmes, et ou A est donn^ par liquation A = y\,

1

,,

(4).

25]

MEMOIRE SUR LES FONCTTONS DOUBLEMENT PERIODIQTJES.

159

II est facile de voir que Ton peut changer la sommation en integration double, et ecrire dmdn

,_. (5)

entre les me'mes limites qu'auparavant, c'est-a-dire que m, n doivent satisfaire aux deux conditions > T, §' < T. Prenons par exemple, pour limite supeYieure, (m2 = m2, w2 = n2), et pour limite infdrieure, m, n sont censes contenir T comme facteur, de maniere qu'ils deviennent infinis avec cette quantity; on suppose aussi m > T, n>T, mais cela est seulement pour la clarte. II devient ndcessaire a ce point de de'finir de plus pres les valeurs de f2, T ; nous ^crirons T = v+v'i (6), ou i = J — 1 ; a>, co', v, v sont des quantites rfelles, telles que mv — m'v ne s'eVanouit pas; c'est la condition pour que £2 : T contienne une partie imaginaire. Avec les coordonnees polaires

=I - (( r-(ftcosdrdd 6' + Tsin6') = IJ (12cos6 + Tsin&f drdd

2 Kn ' ~JJ r-(ftcos 6' + Tsin6') 2 J en repr^sentant par r ce que devient r a la limite supeVieure; 1'int^grale doit etre prise depuis 0 = 0 jusqu'a 0 = 27r. On voit tout de suite que la partie qui contient log T s'eVanouit; done

_ |*

logrd0

Soit a un angle positif plus petit que |TT, tel que tan a = — , on a evidemment

cos0' depuis 0 = — a jusqu'a 0 = + a, ou depuis 0 = IT — a jusqu'a 0 = tr + at, et

±n COS0'

depuis 0 = a. jusqu'a 0 = TT — a, ou depuis 0 = TT + a jusqu'a # = 2TT — a (le signe ambigu, de maniere que r soit toujours positif). En reunissant les parties opposees de l'integrale, on obtient . _ f a (log m — log cos 0) d0 9 f"~a (log n - log sin 0) dd j _ a (12 cos 0 + T sin 0)2 + j a (12 cos 6» + T sin 6>)2 '

160

MEMOIRE SUB LES FONCTIONS DOUBLEMENT PERIODIQUES.

[25

ou, en mettant dans la seconde integrale W — a = a', et \-w — 6 au lieu de 6, A

„ f a (logm-logcos6>)dfl J_ a (XI cos 0 + T sin 0)2

f « (log n - l o g cos 6>)rffl J_v (Tcos 6 + X2 sin 6>)2 '

La premiere intdgrale se reduit a - 2 (log m - log cos 6) T / f l

(entre

+ Ttaifm

les

limites)

2^ fa

_ 4 (log m — log cos a) tan a ~ XI2 - T 2 tan 2 a

j

0

tan26d8 1 2 X ^ - T 2 tan 0

_ 4 (log m — log cos a) tan a 4 ~ XI2 - T2 tan2 a XI2 + T2

L'integrale, dans cette formule " (I + tan2 6) dO _ ftan a dx 2 2 2 2 X2 -T tan 0 J ft T 2*2 ' 0 o s'exprime tout de suite par les logarithmes, mais il faut apporter une attention particuliere a la maniere de determiner quelle valeur doit etre attribuee a, ce logarithme, dans le cas ou la partie reelle en est negative. Je renvoie cette discussion a une Note. Voici le r^sultat auquel j'arrive. En reprdsentant par Lu la valeur principale du logarithme de u, quand il y a une valeur principale (c'est-a-dire quand la partie re"elle de u est positive), et e'crivant L±nU = Lu, ou L {— u) ± iti, selon qu'il y a une valeur principale de log u, ou de log (— u), on a ft*"1"

dx

1

- T V ~ 2OT

r

/ a + TtanaN

" \D,-T

tan a)

en prenant le signe supdrieur pour wv —
+

4

/

i^r

I22TT2'{* ~ * T

H + T tan a\ " il - T tan <x) '

ou, en re'tablissant la valeur de a, a 4mn log s/m2 + n2

+

4

W

/

arCtan

l

n

. fl

7

"*T

mil + nT\ L

^-^^t)

25]

MEMOIBE SUE, LES FONCTIONS DOUBLEMENT PERIODIQUES.

161

Pour avoir la seconde partie de A, il faut changer m, n, O, T en n, m, T, 11. En faisant cela, on change le signe de tot/ — co'v; ainsi il faut e"crire ZT7rj au lieu de X±wj. En reunissant les deux parties, et mettant \ir au lieu de arc tan r

m

+ arc tan —, on n

obtient enfin f

A=

\

O

T

+ T ii±rt

/mO + nT\

T

/nT+mll

r

L± U n mrj" ii U j L " UUT - mn

formule que Ton pourrait rendre plus simple en distinguant les deux cas ou le premier ou le second logarithme a une valeur principale; mais il vaut mieux la laisser sous cette forme. On aurait pu croire que la valeur de A pourrait s'obtenir plus facilement en r^duisant A a la difference de deux integrales ddfinies, les conditions pour les limites e"tant donnees, dans la premiere, par m 2 <m 2 , n 2 < T2, /J,, V, T des quantit^s infiniment petites. On voit tout de suite que V n'est pas pour cela infiniment petit (en effet, sa valeur depend du rapport fi : v et nullement des valeurs absolues de ces quantite"s), et, pour en trouver la valeur, il faudrait se servir de l'analyse prece"dente. Soit, comme exemple, — = GO , A =

IPTT2 { TA

- n)

2m

2TT

o(n + Ti)

T

i i T ' o + Tf

De meme, pour — = oo , r m A =

O2 + T2 f ~ T L±" ^ ~ 1') ~ T 2ir

( _ fli\

2-iri

__2iri

O

En repre"sentant par A', A" ces deux valeurs de A, on a

A'-A" = ±^p = -2S

(10),

ou j'e'cris

C.

21

162

MEMOIRE SUE LES FONCTIONS DOUBLEMENT PERIOUIQUES.

[25

en prenant toujours le signe superieur pour wv' — m'v positif, et le signe inf&ieur pour (ov - (o'v negatif. Soient ug ce que devient u en prenant — = oo, w_s ce que devient cette mtsme fonction en prenant — = oo; on a evidemment

On peut done s'imaginer une fonction V donnee par les Equations =eiex3us

(12).

On verra dans la suite que M8 est analogue a la fonction H (u) de M. Jacobi. II convient cependant, pour la symetrie, de conside'rer, au lieu de ug, la nouvelle fonction U qui vient d'etre definie. Quoique suffisamment donnde par ces Equations, nous allons en chercher encore une autre definition. Pour cela, il faut trouver la valeur de l'integrale definie qui determine A, prise depuis la meme limite infdrieure jusqu'a celle donnee par l'equation mod (m, n) = T. Mettons, comme auparavant, m = r cos 8,

n = r sin 6;

soient aussi flj =
T1 = v — v'i.

Liquation pour la limite sup^rieure se re"duit a r2 (fl cos d + T sin 6) (flj cos 0 + Tj sin 8) = T2, et celle pour la limite inf^rieure, a r=T. On trouve tout de suite l f2T log (Q cos 0 + T sin 6) (n t cos &+T-, sin 6) dd 2

J0

(Xlcos^

"

^ m + Ttanfl) l o g ( n C08 ^ + T s i n 6) ( n > c o s e + T > s i n ^) ( e n t r e * = °>

6=

(13). T - f l tang T t - f l t t a n + O +T tan 6 \O, + T tan 6 fl, + T, tan 0, i+7T

25]

MEMOIBE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES.

163

D'abord f *"

dd

T 1 - 1 2 1 t a n f l _ f i* /

M

Mx

\

+

J _^ O + T tan 6 Oi + T, tan 6 ~ J _hlt \D, + T tan 6 Q,,. + Tx tan #J d0 = 9O f1* 2 2 Jo ft -T tan2^

/,

2fl

T

2il

T > (l+tan'g)1 H2 - T2 tan2 6>J

I* Pi H + T J(1 | 2

A

'

2

7T

On a de meme, en remarquant qu'en changeant 12, T en T, 12, on change le signe de cov — co'v,

, Oj + Tj tan 5

i2j + Txi"

D'un autre T 2 4- O 2

T T -i- Ofl M

iKf

—

cela donne

[

J_

d6

T.-O.tang

7T

T tan 0 Ox + Tx tan 6> T ^ - T j H ( O + T» t-

TO,-^11

fl

(Q

+ Ti

-n + T^TO^T^'

ou enfin f i"

dd

Tj — flj tan 0 _

j _ j T 12 + I tan u I2j + 1 x tan p

wi

12 + Ti

TTTJ

cov — cov

et de m§me T-X2tan0)_ ±iri (12+Ttan6>)2 "fTTTt" en omettant seulement le dernier terme de l'autre int^grale; ce que Ton peut verifier, au reste, en differentiant par rapport a f2, T l'equation TT

21—2

164

MEMOIRE SUR LES FONCTIONS DOCJBLEMENT PERIODIQUES.

[25

On a done, en ajoutant les deux parties qui composent l'integrale,

A +

-- n4nh-+TW^)

<">

II est facile de voir, en e"crivant . _ _ 2m 12 7rTji + ± 12T 12TTi T (T122 - T212)' que cette expression ne change pas de valeur en mettant 121( T1; 12, T au lieu de 12, T, flj, T1; pourvu qu'on change aussi le signe de i; cela doit eVidemment etre ainsi et peut servir de verification. Soit, pour un moment, Wj ce que devient u en prenant pour limite liquation

et supposons que dans la fonction u on ait pour limite liquation mod (m, n) = T. En retenant la valeur de A, qui vient d'etre trouve'e, u= mais aussi

a cause de l'equation, consequence facile des r^sultats pre'ee'dents,

En ^liminant Mj, on obtient, entre w, U, une Equation de la forme

dans laquelle Ti

J

•D —

-a- I

I

"

T

2iri

- I "'

l

Tl7r

T HT 12 Ti ~ -~ OOP I2T -I T (»i/ - »'i ri^ o -r+ W +

ou enfin Q)V -fflJV I

/ - w'u) ~ I2T mod («u' - «B'V)

25]

MEMOIRE SUB, LES PONCTIONS DOUBLEMENT PEEIODIQUES.

165

En rassemblant tous ces resultats, on a le systeme de formules

o _ 7ri

(a>v' — IB'II)

~X2T i l T mod (cov - w'v)' IT (lOV + w'v)

JJ_

mod (wv' — (o'v)' u, ug, u-g des fonctions de la forme

(A) " ( m , n){' mais avec des limites diffe'rentes, savoir: mod (TO, n) = T,

T= <x>, pour la fonction u;

m m? = m2, n2 = n2, m = oo, n = oo, — = oo , pour u$ ; TO2

= m2, n.2 =n 2 ,

n = oo, —=oo, pourM_§.

TO=OO,

A present, il importe de remarquer que la fonction ug est pdriodique a l'egard de O, de la maniere d'un sinus ou d'un cosinus, mais ne Test pas a l'egard de T ; de meme w_g est periodique a l'dgard de T, mais non a l'dgard de il. Quant a U, u, elles ne sont simplement pdriodiques ni a l'egard de il, ni a l'egard de T. Pour de'montrer cela, considdrons, par exemple, l'e'quation ug = x II 1 +

(17)

(m, n).

[m depuis — m jusqu'a m, n depuis — n jusqu'a n, m = o o , n = oo,m— = oo, le systeme (m = 0, n = 0) toujours exclu]; en repre*sentant par u'g ce que devient ue quand on + O au lieu de *, on a dvidemment

D, (TO,

m),

1 (TO + 1,

n)J ' "

l, n) n) '

(TO,

en admettant dans le premier produit la combinaison (TO = 0, n = 0), mais excluant (TO + 1 = 0, n = 0), et excluant l'une et l'autre du second produit. Cette Equation est de la forme

u'e = A'xUL

X

, n)J'

avec les me"mes limites que dans u; done u'g : Ug =

x

(m + 1 , n).

:n

166

MEMOIRE SUB, LBS FONCTIONS DOUBLEMENT PERIOD1QTJES.

[25

ou II se rapporte a la seule variable n qui doit s'e'tendre depuis — n jusqu'a n. Puisqu'ainsi n ne re<joit jamais des valeurs qui soient comparables a, m, chacun de ces produits se reduit a l'unit^, et Ton obtient u'g : ue= A', ou enfin «'<=-Mf

(18),

en posant x= et remarquant que u est fonction impaire de x, ce qui donne A' = - \ . Soit de meme u"g ce que devient w§ en changeant x en x + T.

On a pareillement

x u

U e : Ug = .

^(m, n+l)7 • r ( « , -*)/'

ou II se rapporte a la seule variable m. Mais ici les produits ne se reduisent point a F unite"; en effet,

Mais quand la partie imaginaire de 8 est infinie, on trouve

sin0 = I.(e"-6-*) = ± l e selon que la partie reelle de 0i est positive ou negative. est de signe contraire a cov — ca'v; on a done

iT Or la partie rdelle de -^

(m, n + 1). de m^me,

n

x (m, - n ) /

et de la «"«= ^ . " 6 - ^ ^ ^ equation de la meme forme que celle que Ton obtiendrait en posant

(19),

25]

167

MEMOIRE SUE LES FONCTIONS DOTJBLEMENT PERIODIQUES.

et remarquant que w_g est pe'riodique a l'dgard de T. On voit aussi qu'en sommant par rapport a m, il est permis d'exprimer ug par un produit infini simple, dont chaque facteur est de la forme sin ^ (« + nT), mais qu'on n'arrive pas a un tel rdsultat, en sommant par rapport a n. En effet, liquation fait voir que Ug s'exprime par un tel produit ou chaque terme est de la forme TC

sin = {x + mil), multiple par le facteur exponentiel e~Sx*. On etablit de cette facon l'identite, a un facteur constant pres, de ue a S(u); mais, a present, pour eviter les longueurs, je ne me propose de considdrer aucun de ces produits infinis simples. II faut maintenant, pour le deVeloppement de la thdorie, introduire les trois fonctions qui correspondent au cosinus. Mettant, pour abrdger, (20),

j'ecris

Limites.

mod (TO, n) = T, T' = oo. n)

mod (TO, n) = T.

(m, n) Zx =

l + (m, ==?-=i n)

mod (m, n) = T. mod (TO, fi) = T.

En reprdsentant par y$x, y_g«, &c, ce que deviennent ces fonctions, et prenant pour limites (m2 = m2, « 2 =n 2 ), (m2=m2, n2 = n2), (TO2 = m2, n2 = n2), (m2 = m2, n2 = n2), m . n — = oo pour ygx, &c, — = x pour y_g«, &c, on a, en ge"ne"ral, en mettant ./ au lieu de Tune quelconque des lettres y, g, G, Z,

ou

Jx^e^JeX^e-l^J-tX

(21),

ou Jgx est periodique a l'dgard de fl, et J_gx a l'egard de T. C'est cette Equation remarquable qui d^finit la loi de periodicity des fonctions J, et de laquelle se de"duisent presque toutes les proprietes de ces fonctions.

168

[25

MEMOIRE SUB. LES FONCTIONS DOUBLEMENT PERIODIQUES.

II est clair qu'en changeant entre elles les quantity's 12, T (quantites que nous ddsignerons comme les fonctions completes), on ne change pas les fonctions yx, Zx, tandis que gx se change en Ox, et Gx en gx. On change de cette maniere § en — § et B en — B (par exemple, ygx se change en y-gx, &c). Cette consideration fait voir que toute propriety des fonctions J est double, et donne le moyen le plus facile pour passer d'une proprie'te' quelconque a la propriety correspondante. On tire immediatement des definitions memes les equations (j{-x)

(C)

J

= -yx>

g(-®) = gx> G(-x)

y0 = 0,

gO=l,

= Qx, Z(-x)

G0=l,

= Zx;

Z0 = \;

{ II est facile de ddmontrer, de la meme maniere dont nous avons prouve" la periodicite de Jgx a l'egard de 12, que ces fonctions se changent l'une en l'autre, a, un facteur constant pres, en changeant x en te + ^Sl. Faisant done attention a l'dquation qui lie ensemble Jx et Jgx, on obtient le systeme de formules Byx,

(22)

Pour determiner A, B, G, D, posons = 0 ou x = — En e"crivant, pour abr^ger, (D) on trouve

(23)

d'ou Ton deduit cette equation de condition, (24).

On a de me'me le systeme Zx, yx,

(25).

25]

169

MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES.

De la, si irYi

ce qui donne (E)

log q log q1 = - -TT2,

il resulte

(26);

d'ou nous tirons l'equation de condition (27). En posant s; = | T dans liquation pour y(x + \ 12) et a; = ^ O ), on obtient aussi

dans l'equation pour (28),

et Ton deduirait cette mekme equation, ou une Equation equivalente, des autres formules, de maniere qu'on ne peut pas trouver d'autres Equations de condition. Enfin, en rapprochant ces systemes, on obtient y (x + £ ft + i T ) = e ^ B"gx,

(29),

Z (« + \ ft + 1 T ) = el avec les equations suivantes pour les coefficients,

:y(i«).

(30).

: y(i T).J

En rassemblant les equations entre

, /(JT), on a

(31),

C.

22

170

[25

MEMOIRE SUB. LES FONCTIONS DOUBLEMENT PERIODIQUES.

d'ou Ton conclut les formules B"C" : A"D" = B'D' : A'C = CD : AB = - 1 ,

A'B' : CD' = -A"B" : CD" = - y2 (£ T) : A"C" : B"D" = -A' C : B' D1 = A D : B C --A'B' : B' C =

,-

.(32),

') dont nous aurons bientdt besoin. On peut passer maintenant a ce systeme general de formules, ou j'ecris (—)'" au lieu de ( - l ) m :

la, comme toujours, (TO, — n) = mil - wT ; y {x + (TO,TO)}= ( - ) m + n &yx, g [x+(m, w ) } = ( - ) m

0ga?,

' {x + (TO, n)} = ) =

(~)'lm

"e8*1"1'

y {x + (TO, «)} = ( - )

*}, m+

g {a; + (w, n)} =( — )

m

4

'

5"

" ®Agx, <£>Byx,

G{x + (TO, n)} = ( - ) »

$az«,

y {a; + (TO, n)} = ( - ) m + n VA'Gx, g [x + {m, n ) } = ( - ) m

Z[sc+(m,

n)}=

n)}=(-)m+nXA"Zx,

g{*+(TO, n)} = ( - ) » XB"Gx, G{x+ (TO, n)}=(Z{x+

(m, n)} =

)n

XC'gx, XU'yx.

25]

MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQTJES. Soit

171

x=0:

y (m, ») = g (m,

0,

/(»», n) = (-y»+»e0,

»») = (-)« e0,

G(m, n) = (-)« ©0) £(m,n) =

@0;

<|) = ( _ \mn+jtm „ — in

y (m, w ) = ( - ) ™ + » * g(mi, ??)=

0,

(G) y (m, n) = ( - )TO+W ^ 0 4 ' , g(m, n) = ( - ) » G (m, n) =

VoB', 0,

0' (m, n) = ( - ) »

Z (m, n) =

j(m,

n) = (-)•»+»Xoii",

g(w»,n) = ( - ) » X0B", yZ{m,n)=

0,

Z' (m, n) = X0D".

On a de suite, en prenant les diffe"rentielles des logarithmes des fonctions Jx,

g' a; : gx = —

(H) G'x

•.Gx=-

{x- (m, ifi)}-1,

{x-(m, n)}~\ (B est le coefficient de a?, dans l'exponentielle e^Bx2 des Equations (B); il n'y a pas a craindre de le confondre avec le B qui entre dans les equations pour y (m, n), &c.: c'est par hasard que j'ai pris deux fois la meme lettre.) 22—2

172

MEMOIRE SUR I/ES FONCTIONS DOUBLEMENT PERIODIQUES.

[25

Re'duisons en fractions simples la fonction

gx Ox : yx Zx. En ecrivant n)}-1 + M [x - (m, n)}'1],

gxGx : yxZx = ^[L{x-(m, on obtient L=g(m,

n) G(m, n) : y'(m, n)Z (m, n) = 1,

M = g (m, n) G (m, n) : y (m, n) Z' (m, n) = B"C" - A"D" = 1. Done , n)}'1 — 2 {« — (m, n)}~1,

gx Gx : yx Zx =

e'est-a-dire gx Gx : yx Zx — (y'x : yx) — (Z'x : Zx).

(Nous allons bientot justifier, au moyen d'un thdoreme de M. Cauchy, l'emploi de cette methode de decomposition en fractions simples qui ne s'applique pas toujours aux fonctions transcendantes.) On obtient de meme gx Zx : yx Gx = (y'x : yx) — (G'x : Gx), GxZx : yx gx= (y'x : yx) — (g'x : gx), : gx Gx = (g'x : gx)—(G'x : Gx),

(I) 2

e yxgx : GxZx = (G'x : Gx)-(Z'x

: Zx),

d'yxGx : Zx yx = (Z'x : Zx) — (y' x : y x), en mettant, pour abreger, y(iT)

: _i c'

(J)

Done, en eliminant les fonctions derivees, g'x -G2x = Z*x - g*x =

c^x,

i . 1 J'ai &:rit - au lieu de - pour me eonformer a la notation d'Abel; il est a peine neoessaire de remarquer que c, e sont, en general, l'une et l'autre des quantites imaginaires. 1

25]

MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES.

173

ce qui donne, en ajoutant, (K)

6 2 =e 2 +c 2 , ou 6=s/e 2 + c2,

en nous retiendrons desormais la lettre b dans cette signification. 2

Puis

2

( g x = Z x - cytv,

(L)

-i

2

Soient maintenant (x =jx : Zx, (M)

•! fx = gx : Zx, [Fx=Ox:

ZX.

On obtient r Px = 1 - cWx, Mr. l i ^ * = 1 +e2<jy'x;

(N)

(
(0)

]/'« = -

Ces Equations sont precisement les equations fondaraentales d'Abel (CEuvres, t. i., p. 143 [Ed. 2, p. 268]), et il en de'duit

1 + e2c22x2y

'

qui sont les forraules connues pour l'addition des fonctions elliptiques. En effet, on peut dcrire l'dquation

x ~fx Fx sous la forme V (1 — c 2 r 2 #) (1 + e2d>2«) ou, en mettant y = x, (Q)

«=

.

"

;

Jo 7(i-cy)(i + e y)'

174

MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES.

[25

on peut done de ce point supposer connues toutes les propriety's des fonctions elliptiques. On a, par exemple,

et de la i

,

dy qui determinent fl, T en fonction de c, e. II parait au premier abord que T, H soient des fonctions parfaitement determine'es de c, e. J'ai des raisons pour croire que cela n'est pas precise'ment le cas, et que la question admettrait des deVeloppements interessants; mais je reserve ce sujet pour une autre occasion. On peut, a l'aide des Equations entre y(^fl), &c, et les quantity's c, e, exprimer cette suite de fonctions en termes de c, e. On deduit: 'y QsQ,) = b-lc-iq1-l,

y (^T) = * - * e - ^ - * ;

(S)

Z(iO)=6-*c*gr*,

^(iT)=i

et de la

(T) v '

^=bic~iqr*

,

C' = -ibieiq~> ,

D = b~l cl qr* , Jy = b~i el
G" = (-)*c~*elqr D" = - (-)

qu'on doit introduire dans toutes les formules ou entrent les quantite's A, B, &c. Voici le theoreme de M. Cauchy (Exercices de Maihdmatiques, tome II. page 289): " Si, en attribuant au module r de la variable z — r (cos p + i siup) des valeurs infiniment grandes, on peut les choisir de maniere que les deux fonctions 2

'

2z

deviennent sensiblement nulles, quel que soit Tangle p, ou du moins que chacune de ces fonctions reste toujours finie ou infiniment petite, et ne cesse d'etre infiniment

25]

MEMOIRE SUB LES FONCTIONS DOUBLEMENT PERIODIQTTES.

175

petite, en demeurant finie, que dans le voisinage de certaines valeurs particulieres de p, on aura !— Z

pourvu qu'on re"duise le residu integral a sa valeur principale." On se rappelle que cela veut dire qu'en supposant ces conditions satisfaites, la fonction fx peut s'exprimer de la maniere ordinaire comme la somme d'une suite de fractions simples, mais qu'il faut e'tendre d'abord la sommation aux racines de liquation

i = 0. fx dont les modules sont infeYieurs a une certaine limite qu'on fait alors infinie. Soit, par exemple,

ou Tx, T^x ne contiennent que des facteurs qui sont des puissances entieres des fonctions jx, gx, Goc, Zx. En supposant que r n'est d'aucune des formes mod (m, n), mod (m, n), mod (m, n), mod (m, n), mais, d'ailleurs, uue quantity infinie quelconque, on peut toujours ecrire z = r (cos p + i sinp) = (m, n) + 0 ou 6 est une quantite m, n sont des entiers jusqu'a oo, avec Tangle &c.; soit aussi X = 8j —

(34),

finie telle qu'aucune des fonctions y^, gd, GO, Z6 ne s'evanouit, dont l'un au moins est infini, mais qui varient depuis — oo p. Soit & le degre de Tx, H, celui de Txx a l'egard de yx, 8 ; on a, en general, une Equation de cette forme, fe

q^

qh\n» eIm+Jn

]?

ou F est fini. En supposant done que la partie reelle de a(m, ny-\§Wm?+\§T3rii

(36)

est negative quelles que soient les valeurs de m, n, on a

et de m^me

f*-f(-z)_Q 2z Si, au contraire, cette partie reelle est positive, ces trois fonctions sont toujours infinies. U y a cependant un cas particulier a, consideVer, savoir, celui ou cette partie rdelle se re"duit a Pm? ou Qn2, P, Q e"tant des quantit^s positives. II faut ici que le

176

MEMOIRE STJR LES FONCTIONS DOUBLEMENT PERIODIQUES.

[25

coefficient de n ou de m dans la partie resile de Im + Jn s'evanouisse. parties re'elles s'eVanouissent entierement, ce qui ne peut arriver que pour a=0,

6 = 0,

Enfin, si les

\ = 0,

fx est fini. On a done pour fx fonction impaire, 2

~ '


(la seconde Equation a cause du de'nominateur infini z). II est cependant certain que, dans plusieurs cas pour lesquels fx est fonction paire, on peut re'duire fx en une suite de fractions simples: par exemple, jx : gx est une fonction impaire que Fon peut, par ce qui precede, deVelopper en suite de fractions simples; en e'erivant x + %T au lieu de x, on ddduit un pareil developpement pour Gx : Zx qui est fonction paire. Remarquons que quand la partie reelle de a(m,

rif - AgH2m2 + XGVn"

est negative pour toute valeur de m ou n, la suite pour fx est toujours convergente. En effet, les nume'rateurs des fractions simples contiennent ce facteur de fz,

qui s'evanouit pour les valeurs infiniment grandes de m, n. Dans le cas ou la partie reelle est positive, le developpement ne peut jamais etre vrai; je crois qu'en general il est vrai, dans les cas limites, quand la suite que Ton obtient est convergente. II y a des exceptions cependant; on en verra une en deVeloppant (ffx. Avant de passer aux exemples, developpons la condition pour que la partie reelle en question soit toujours negative. En supposant

a = h + ki, on obtient pour cette quantite l'expression \TT mod (eov — (o'v)\ "" m?\h (tu2 - «'2) - 2kwco' -' v2 + 1 / 2 j. / a

'a\

OJ

h (u2 — u 2 ) — 2kvv

/

(37),

A,7r m o d ( w v — <» v) 2

—-.2

O) + ft)

+ 2mn {h(wv — m'v) — k(cov +
qui doit toujours rester negative. h? + &2 + 7-r-—,. . (a) 2 + ft)2)

(OT2

X 2 77 2

,

Cela donne, apres quelques reductions, la condition

,fr-^V,—-,

r\ \(a>v - ft)V) /* + (wv + IO'V) k]

+TO-2 ) m o d (wv — <» i/)

v

v

\

'

\...m\ I

25]

MEMOIRE SUB LES FONCTIONS DOTJBLEMENT PERIODIQUES.

Les valeurs

177

h = 0, k = 0

satisfont a cette condition, laquelle du reste (en considerant h, k comme les coordonnees d'un point) est satisfaite pour tout point situe en dedans d'un certain cercle qui inclut l'origine, et dont on aurait l'equation en remplacant le signe < par le signe = dans la formule (38). On obtient de cette facon une grande varie'te' de formules particulieres. exemple celles-ci:

Par

(m, n)*+b (m, i ; (m,

(XJ)< - (m, n)

dans lesquelles, pour trouver les limites de a, il faut faire X = 1 dans la formule (38). On a ensuite ce systeme de formules sans exponentielles, (,

: yx =

2 [(-)"

[x - (m, n)}-1],

: yx =

S [(-)m+n

[cc - (m, w)}"1] ;

yx : gx = — 1 ' Gx : gx = -

cr1 2 [(-)«+» [x - (m,

Zx : gx = —

b~x S [(—)m

(V)

ya? : era;

: fe =

-^ ~i [ ( - ) m

{x - (m, {^ - (TO, n)}-1],

![(-)"+» {*-(m, n)}-1],

: Gx = S [(-) TO+n [x - (m, n)}"1], gx : ^« =

S [(-)•»

{a; - (m, a)}"1],

: ^x =

"

{* - (m, n)}"1],

dont quelques-unes ont ete donndes par Abel. II faut remarquer que celles de ces - equations ou la fonction est impaire sont justifies par le thdoreme de M. Cauchy, et que les autres se deduisent de celles-ci. On peut trouver de m§me le developpement des fonctions 1 Gx yx gx yx gx' " ' ya: gx''" ' Zx Gx c. 23

178

[25

MEMOIRE SUR LES FONOTIONS DOUBLEMENT PERIODIQUES.

(celles-ci, qui sont toutes impaires, ont e*te deja conside're'es; nous venons de faire voir que le deVeloppement est admissible); Zx yxgxGx'"''

1 yxgxGx' '" '

chacune, except^ celles de la suite rJ^7^,

., ° ,

1 yxgxZx''" '

multiplied par un facteur exponentiel

Par exemple, 1 {« - (m,

(W)

{* - (m,

Mais on est conduit a un resultat beaucoup plus important en considerant, par exemple, le deVeloppement de (fy'x. Cette fonction contient non-seulement une suite de fractions simples, mais aussi un terme constant; nous ^crirons

{x-(m, n)}-2 + M{x-(m, n)}-1]

(39).

Pour determiner L, M de la maniere la plus simple, changeons x en x+ | i l + JT. En faisant attention a la valeur de <\>x = yx : Zx, on obtient - e~ 2

-* = J +

{x - (m, n)}-* + M{x- (m, n)}'1];

de la L = -

{* - (m, n)Y ^-[{*-(m,

],

« = (m. «),

ou enfin L = - e~2 c~2 x2

, pour « = 0,

ce qui donne Z = -e-2c-2,

M=0,

<jy!x = J- e"2 c~2 S {* - (m, n)}-2.

(40).

En integrant deux fois fx

rx

x-(m, n)\

dx Jo

Jo

(41),

25]

MEMOIRE SUE. LES FONCTIONS DOUBLEMENT PERIODIQUES.

179

ou, a cause de log Zx = — \Ba? + S log [x — (m, n)}, f d x [ ' <$?xdx = \Ix* + e-Jie-*\Q>gZx

(42),

Jo J o '

en mettant, pour abre"ger, 1 = J -e~*c^B (on n'a pas ajoute de constante arbitraire, parce que Zx est fonction paire de x qui se reduit a l'unite' pour x = 0). Puis, on a

(43).

l>K De la il est facile de determiner la valeur de I. rx

§tx = I 2x dx, Jo

Soit, pour un moment, rx

(p^x = I (ptx dx. Jo

Puisque

^(x + n)-(f>'x=0, on a „ (x + O) - (pux = xfat llais de liquation tfx - p (XI - x) = 0 on d^duit ou, en faisant x = \ fl, c'est-a-dire

n et de la

Mais on a deja. Z(x + O) = done enfin

c'est-a-dire (44)

23—2

-

180

MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES.

[25

Soit, pour abre"ger,

dy'x dx ii

(45);

J0

on a cette Equation, Jo

Zx = s

(X)

Jo

>

qui exprime la fonction Zx au moyen de x. C'est, en effet, la formule remarquable de M. Jacobi, que nous avons citee dans l'introduction de ce Me'moire. En changeant seulement la notation, on a, d'apres M. Jacobi, 4 (
= 'H^*/^

j0

*

~ V <* ~ ffl>} dx

,...

(46),

ZAafa

ou, ce qui est la me'me chose,

De la, en multipliant par e2c2, et faisant attention a la valeur de Zx, Z' (x + a) _ Z' (x - a) _ 2Z'a _ 2escs afa Fa if>•x

Z(x+a)

Z(x-a)

Za

(49).

Ecrivant dans cette Equation a, x au lieu de x, a, et ajoutant, on obtient Z'x

Z'a Z'(x + a)

(oO),

En integrant la mSme equation par rapport a a, logZ(x + a) + hgZ(x-a)-2IogZx-2hgZa c'est-a-dire ^(a; + a)^(«-a)=^ 2 a;Z 2 a(l+e 2 c 2 ^ 2 a^a;)

(52),

ou, ce qui est la me'me chose, (53), de laquelle on deduit facilement, en Ecrivant x + ^il, x + ^T, * + | H + | T au lieu de x, les equations compldmentaires (Y)

(y (x + a)y (x-a) = y'x Z*a - y \g(x+a)g(ne-a) =

25]

MEMOIRE SUR LES FONCTIONS DOTJBLEMENT PERIODIQUES.

181

Quoiqu'elle ne soit pas liee tres-etroitement avec, la theorie actuelle, on peut aj outer ici cette autre formule de M. Jacobi, qu'il obtient en integrant par rapport a x, au lieu de a, (x-a)

M

+

Z'a

,,., (4)

au moyen de laquelle il deduit des formules pour l'addition des arguments ou des parametres des fonctions de la troisieme espece. On trouve aussi, dans les Fund. Nova, i

r

i

j / j -J.

J

1^

4.-

/An\

•

Z(x-a)Z(y—a)Z(x+y

+ a)

quelques formules deduites de 1 equation (49) pour expnmer ~^. ( „;' -r~ - (• 1 ^ X / J . a. Z {x + a)Z (y + a)Z (x + y — a) au moyen de la fonction ; il serait facile de deduire des Equations semblables pour les autres fonctions y, g, G.

Note sur une integrale definie. Soient kx, k des quantites replies dont la premiere est la plus grande; H — a> + a>'i, T = v + v'i des quantites quelconques, telles que
(55).

L'integrale u a toujours une valeur finie et determine'e, puisque le ddnominateur ne devient jamais ze"ro. On a evidemment

L

[fl + T ^ f l + T ^ ]

(56)

'

et l'int^grale indefinie est

il faut passer de la a l'integrale definie, entre les limites 0, oo. Soit

II est facile de voir qu'en faisant abstraction d'un de*nominateur toujours positif, Ax est une fonction du second degre" en x, et Bx se rdduit a la quantity constante (a>v — (o'v) (h — k). Le signe de Bx est done toujours le meme que celui de cov — oo'v; quant a celui de Ax, puisque dvidemment ^ . ^ = 1 , il est clair que si Ao est positif, Ax reste toujours positif, ou change deux fois de signe quand * passe de 0 a oo. Si, au contraire, Ao est negatif, Ax est ne"gatif depuis x = 0 jusqu'a une certaine valeur positive de x, x= a, et positif depuis x = a jusqu'a x = oo. ConsideYons d'abord ce dernier cas. En representant par Lx la valeur principale de log x (cela suppose que la partie re"elle de x est positive), on obtient cette valeur de u, U = ^1J

fI —fl^ L O

.

/^T

\ — -zz L \ — ^

^ f——;

^ \ — f^L

182

MEMOIRE SUB, LES FONOTIONS DOUBLEMENT PER1ODIQUES.

oil e est suppose une quantite infiniment petite positive. se reduit a /

,

-Da

,

[25

La somme des derniers termes Aa A

-arctan-j— +arctan ^ 1

/-n\

(o9),

oil, comme a l'ordinaire, arc tan x doit 6tre situe" entre les limites + fyir. Dans cette foraiule, -4o_e est une quantite" infiniment petite et negative; Aa+t est une quantity infiniment petite et positive; done, quand Ba est negatif, les arcs se reduisent a \ir, — \TT, et si Ba est positif, a — ^w , ^TT . On a done i . Tm

. (60)

'

ou il faut se servir du signe supe"rieur ou inf^rieur selon que wv — a>'v est positif ou negatif. Dans le cas ou Ao est positif, si Ax reste toujours positif, on obtient tout de suite

Si Ax change deux fois de signe, par exemple pour x = a et x = §, il faut introduire les corrections ^ (— arc tan--j- 5 - + arc tan-p 2 -) + ~ ( — arc tan -~- + arc tan -r-^ qui se ddtruisent l'une Fautre, en sorte que Ton a le meme r^sultat que si Ax etait toujours positif. En se servant de la notation employee dans le Me"moire, on a dans tous les cas + TfeA

J

ou le signe se determine d'apres celui de cov — ov'v. En particulier, fk

_

|-tan«

i0

^

dx

_

_ 1 .

l

G'-TV~2I2T

Je ne sais pas si Ton a cherche" avant moi la valeur exacte de cette integrate de'finie, qui est cependant la plus simple qu'on puisse imaginer, et qui devrait, je crois, trouver place dans les livres ele'mentaires. NOTA. Une partie de ces recherches a dtd deja imprime'e dans le Cambridge Mathematical Journal [24] ; je me suis borne* au cas ou O, T sont de la forme a>, vi, ce qui simpline beaucoup la determination des int^grales de'finies doubles; mais la forme geneYale des rdsultats en est tres-peu affectde.

26]

183

26. MEMOIRE SUR LES COURBES DU TROISIEME ORDRE. [From the Journal de Mathematiques Pures et Appliquees (Liouville), tome ix. (1844), pp. 285—293.] d'abord la surface du troisieme ordre qui passe par les six aretes d'un tetraedre quelconque. Cette surface sera touchee selon chaque arete par un seul plan; je dis que: "les plans tangents selon les aretes oppose"es se rencontrent en trois droites qui sont dans le m&me plan, et chacune de ces droites est situde entierement sur la surface." En effet, si nous representons par P = 0, Q = 0, R = 0, 8 = 0, les equations des quatre plans du tetraedre, et par a, §, y, 8 des constantes arbitraires, l'equation de la surface ne peut avoir que la forme <xQR8 + SPRS + yPQS + BPQR = 0. Soient ^ , ©, JSJL, %fa ce que deviennent les quantite's P, Q, R, S, quand on change les coordonne'es x, y, z en de nouvelles variables f, 77, £; l'equation du plan tangent se reduit facilement a la forme CONSID^RONS

( $ - P) (§RS + jQS + BQR) + ( © - Q) (aRS + yPS + BPR) + ( H - R) (aQS + §PS 4 BPQ) + (ft - S) (aQR + SPR + yPQ) = 0. Soient P = 0, Q = 0 ; cette equation devient
=o,

184

MEMOIRE SUE, LES COURBES DU TROISIEME ORDRE.

[26

On conclut de la premiere, que la droite d'intersection des deux plans tangents est situee sur la surface; et de la seconde, que cette droite est dans un m6me plan, quelles que soient les deux aretes opposees par lesquelles on a mene" les deux plans tangents. Ainsi le the'oreme est de'montre. En conside"rant une section quelconque de cette surface, ou meme en supposant que P, Q, R, S ne contiennent que deux variables x, y, nous avons ce the'oreme: "Etant donnee une courbe du troisieme ordre qui passe par les six points d'intersection de quatre droites, les tangentes a la courbe en ces six points se rencontrent deux a deux en trois points qui sont les points d'intersection de la courbe par une droite. Les tangentes qui doivent etre prises ensemble sont les tangentes en deux points A, A', tels que A est intersection de deux des quatre lignes donnees, et A' l'intersection des deux autres lignes, points que Ton peut nommer opposSs. " De meme, si une courbe du troisieme ordre passe par cinq de ces points, de telle maniere que l'intersection des tangentes a la courbe en deux points opposes soit situe'e sur la courbe, la courbe passe par le sixieme point, et par les points d'intersection des tangentes mendes par les deux autres paires de points opposes." A present, posons une courbe quelconque du troisieme ordre et deux points A, A' sur la courbe, tels que les tangentes en ces deux points se rencontrent sur la courbe (cela suppose que la courbe est de la sixieme ou quatrieme classe, et non pas de la troisieme). Un autre point JB sur la courbe e"tant pris a volontd, les droites AB, A'B rencontrent la courbe en H et h; et les droites Ah, A'H se rencontrent en un point B' situe sur la courbe. Les tangentes en B, B', et de meme les tangentes en H, h, se rencontrent sur la courbe en deux points qui sont en ligne droite avec le point d'intersection des tangentes en A et A'. On peut dire que les deux points B, B', ou les deux points H, h, sont une paire de points correspondante a la paire A, A'. Les deux paires B, B' et H, h sont evidemment correspondantes l'une a l'autre, et, de plus, A, A' correspond a ces deux paires, de la meme maniere que H, h correspond a A, A', et JB, B'; de sorte que Ton peut dire que les trois paires A, A'; B, B'; H, h sont supplementaires l'une aux deux autres. Soit C, C une autre paire de points correspondante a A, A'; je dis que B, B' et G, C sont des paires correspondantes l'une a, l'autre; et, de plus, si d'un point P quelconque de la courbe, Von mene des lignes aux points A, A', B, B', C, G', ces lignes forment un faisceau en involution.

En effet, considerons six points quelconques A, A', B, B', C, C dans le meme plan, et representons par f, g, h les points d'intersection de BC et B'G, CA' et C'A, AB' et A'B, et par F, G, H les points d'intersection de GB et C'B', CA et C'A', AB et A'B'. Nous allons faire voir que le lieu d'un point P qui se meut de telle maniere que les lignes menses aux points A, B, C, A', B', G' forment toujours un faisceau en involution, est une courbe du troisieme ordre qui passe par ces six points, et aussi par les points / g, h, F, G, H.

26]

MEMOIRE SUE, LES COT7RBES DU TROISIEME ORDRE.

185

Soient a, a', S, §', y, 7' les tangentes trigonome'triques des inclinaisons des lignes PA, PA', PB, PB', PC, PC, sur une ligne fixe quelconque que Ton peut prendre pour l'axe des x. Pour que ces lignes forment un faisceau en involution, nous devrions avoir la relation cut (g + g' - y - y') + g§' (y + y' - a - at) + 77' (a + at - g - g') = 0, ou, ce qui revient a la me'me chose, l'une quelconque des quatre equations

Soient xP, yP, xA, yA, etc., les coordonnees de P, A, etc. , _

vOp

0Cj^

SGp ^~ Cuj^'

\ p ~~~ *^Aj \ P ~~

B')

en reprdsentant par (PAB'), etc., les quantit^s telles que

L'equation de la courbe peut done se mettre sous l'une quelconque des quatre formes

PAB' . PBC . PCA' + PA'B . PB'C . PC A = 0, PA'B'. PBC . PCA + PAB . PB'C . PCA' = 0, PAB . PB'C . PCA' + PA'B' . PBC . PCA = 0, PA'B . PB'C . PCA + PAB' . PBC . PCA' = 0, qui sont du troisieme ordre. La premiere fait voir que la courbe cherchee passe par les neuf points A, B, C, A', B', C, f, g, h. La troisieme ou la quatrieme fait voir qu'elle passe de plus par le point F; la quatrieme ou la seconde, qu'elle passe par le point G; la seconde ou la troisieme, qu'elle passe par le point H. Ainsi la courbe passe par les douze points A, B, C, A', B', C, f, g, h, F, G, H. On peut remarquer qu'une courbe du troisieme ordre qui passe par dix quelconques de ces points, ou meme par neuf quelconques, pourvu que nous exceptions les combinaisons de A, B, C, A', B', C, avec / , g, h, ou / , G, H, ou F, g, H, ou F, G, h, ne peut £tre que la courbe que nous venons de trouver. On peut aussi remarquer en passant que si A, A', B, B', C, C sont les points d'intersection de quatre droites, l'e'quation cidevant trouve'e est satisfaite identiquement, de sorte que la position du point P est absolument arbitraire. Cela e'tant connu, je ne m'arre'te pas pour le demontrer. Eevenons au cas d'une courbe donnee, avec trois paires de points A, A', B, B', C, C, comme auparavant, tels que B, B' et C, C sont des paires correspondantes a A, A'. Les C. 24

186

MEMOIRE SUE LES COURBES DU TROISIEME ORDRE.

[26

points A, B, C, A', B', C, 0, g, H, h sont situe"s sur la courbe; ainsi F, f sont aussi sur la courbe (c'est-a-dire que B, B' et G, C sont des paires correspondantes), et les lignes menses par un point P quelconque de la courbe et les points A, B, G, A', B', C forment un faisceau en involution: theoreme ci-devant e'nonce'. Considerons une courbe du troisieme ordre, de la quatrieme classe (c'est-a-dire, telle que d'un point quelconque on ne peut lui mener que quatre tangentes). D'un point sur la courbe, independamment de la tangente en ce point, on ne peut mener que deux tangentes. Prenons les deux points K, L sur la courbe, et menons les tangentes KA, KA', LB, LB' touchant la courbe en A, A', B, B'. II est clair qu'en ce cas A, A' et B, B' sont des paires correspondantes de points. Mais le cas general ou la courbe est de la sixieme classe est moins simple. Considerons, en effet, pour une telle courbe, les huit points Alt A2, As, A4 et Bu B2, B3, B4 de contact des tangentes menees par les deux points K et L. En choisissant A, A' et B de quelque maniere que ce soit, parmi les points Alt A2, A3, A4 et Blt B2, B3, B4 respectivement, le point B', qui doit entrer dans cette derniere se"rie, ne peut pas etre choisi a, volonte", mais est parfaitement determine'. En designant convenablement les points B, on peut toujours supposer que les paires correspondantes soient AjA2 ou A3A4 avec BA ou B3B4; A^s ou A2A4 avec BXB3 ou B2B4; AXA4 ou A^A3 avec BA ou BJZ3- Cela suppose que

AA,

A2B2, A3BS, AJBt;

AA,

A A,

AA,

A A,

AA,

AA,

AA,

A A;

se rencontrent, chaque systeme, dans le m§me point. dence la correlation de ces huit points.

A A,

A A;

AA,

II faut expliquer avec plus d'evi-

Imaginons les six points A, A'; B, B'; G,G', tels que AA', BB', GG' se rencontrent dans le meme point. Formons, comme auparavant, le systeme de points f, g, h, F, G, H. Les proprie'te's de ce systeme de douze points sont tres-nombreuses. Non-seulement F, G, H; F, g, h; f, G, h; f, g, H sont en ligne droite; mais, en outre, les trois droites de chacun des onze systemes que voici se rencontrent dans le meme point: Af , Bg ,

Ch ;

A'f,

B'g,

G'h ;

AA',

Gg , Hh ;

BB',

Hh,

Ff

A'f,

BG,

CH ;

B'g,

GH,

CH;

Bg , CH,

Af

, B'G,

;

GG', Ff,

Gg ;

AF;

Ch,

AF,

BG ;

A'F;

Ch,

A'F,

B'G.

(Cela est connu, je crois; au reste, pour le ddmontrer, consideVons un parallelipipede, tel que gf, GF, AB, A'F en soient quatre aretes paralleles, les autres aretes paralleles etant gA, A'G, fB, B'F; gA', A'G, Bf FF. Soient H le point de rencontre, a l'infini, du premier systeme d'aretes paralleles; C et C les points de rencontre, a l'infini, des aretes des second et troisieme systemes respectivement; h le point de rencontre des quatre diagonales du parallelipipede; faisons la perspective de ce systeme, de"signant chaque point de la projection par la meme lettre: Ton voit sans peine que la figure plane, ainsi obtenue, a les proprietes en question.)

26]

MEMOIRE SUR LES COURBES DU TROISIEME ORDRE.

187

A present, en examiaant la figure, on voit qu'il est permis de prendre pour Alt A2, As, A4 les points A, A', F, f, et pour B1} B2, Bs, B4 les points B, B', G, g, pour que les systemes Alt A2, A3, J. 4; Blt B2, B3, j?4 aient la correlation ci-devant trouvee. Le systeme C, C, H, h est suppldmentaire a ces deux-ci. Toutes les proprie'te's que je proprie"te correspondante que Ton Nous avons ainsi des propri^tes classe et du sixieme ou quatrieme

viens de trouver sont telles qu'il y a a chacune une peut obtenir par la the'orie des polaires rdciproques. non moms intdressantes des courbes de la troisieme ordre.

En prenant pour la courbe du troisieme ordre l'ensemble d'une section conique et d'une droite, pour que les tangentes en A, A' se rencontrent sur la courbe, il faut que A, A' soient situdes sur la section conique de telle maniere que AA' passe par le p61e de la droite a l'e"gard de la conique. Alors B, B'; C, C etant pris de la meme maniere, BC, B'G et BC, B'C se rencontrent sur la droite. Les lignes tirees d'un point P quelconque de la conique, ou de la droite a A, B, C; A', B', C, forment un faisceau en involution. Le premier th^oreme et la premiere partie du second sont tres-bien connus; je ne sais s'il en est de meme de la derniere partie de ce th^oreme.

ADDITION. La droite PP', menee par deux points P, P' d'une courbe du troisieme ordre, qui correspondent toujours a une poire correspondante donne'e de points A, A', est toujours tangente a une certaine courbe de la troisieme classe.

Pour demontrer ce thdoreme, imaginons une paire fixe B, B' de points qui corresponde a la paire donne'e A, A'. Soient P=0, Q = 0, -R = 0, 8 = 0 les Equations des lignes qui joignent A, A' avec B, B'; liquation de la courbe donne'e du troisieme ordre peut se mettre sous la forme

On peut toujours supposer, sans perte de ge'ne'ralite', que l'dquation 0

(2)

soit identiquement vraie, car chaque equation de la forme P = 0 peut 6tre censee contenir une constante arbitraire qui en multiplie tous les termes. Done, en faisant P + R = 6,

P = B-R,

on peut toujours supposer Q + S = -e,

Q=

-d-S, 24—2

188

MEMOIRE SUR LES COURBES DTT TROISIEME ORDRE.

[26

et l'e'quation de la courbe se transforme en 9-R

0+8

R

+

w

S

ce que Ton peut e"crire aussi sous la forme

. (0 + XR) (8 + fiR)+

... l

( - 6 + vS) ( - 0 + PS)

>'

en determinant convenablement les constantes A., /tt, v, p. En effet, en r^duisant, les deux Equations deviennent identiques au moyen des quatre conditions B — — kX/ji,

7 = — kvp, ~\

B-§ = k\/M(vp + v + p), I

(5),

7 — a = kvp (Xfi + X + yu.),; ou k est une quantite arbitraire, de maniere que des quantit^s \ seule qui peut etre prise a volontd.

p., v, p, il y en a une

De l'equation (4) Ton deduit tout de suite cette autre forme,

Rj et de la nous voyons que les points donnas par les deux systemes d'equations

correspondent toujours l'un a l'autre et aux points donne"s par les deux systemes

<"=0' (61 = 0,

fi

=°H

(8 ),

8=0),)

c'est-a-dire aux points A, A'. On trouve assez facilement pour l'equation de la droite menee par les points determines par les deux systemes (7), points que Ton peut prendre pour P et P', (fiv - pX) 6 + Xp, (v - p) R + vp (X - fi) S = 0 ou

C0 + AR + BS = O

(9), (10),

en faisant pC = fiv — pX PA

,}

= \p(v-p),l pB = vp (X - p). J

(11).

26]

MEMOIRE SUR LES COURBES DU TROISIEME ORDRE.

189

Eliminons des equations (5) et (11) les cinq quantity's X, fi, v, p, p. Pour operer de la maniere la plus elegante, formons d'abord les equations identiques 1

— pX — vp (X — //,)] (y —

• — pX -f Xu (v — p)] (X — _

'

.(12),

\fi (v — pf — vp (X — ixf = (fiv — pX) (Xv - up), ce qui donne

A(C-B)(y-a)+B(C + A)($-t k = - \fivp (v\ - up) (2C + A- B),

.(13),

- £2S = - - \/ivp (v\ - fip) G, et de la

"'

''

N

'

/v> § )

+ (2C + A - B) (A*y - B2B)

H =0

(14),

)

ou, toute reduction faite,

A (A + C) (A + C - By) + B(G - B) (A + B - C8)\ _ - AC(C - Ba) - BG (G + AS) ) ~ et puisque cette Equation est du troisieme ordre a Fegard des quantites A. B, G, il est Evident que la ligne donnee par liquation (10) est toujours tangente a une certaine courbe de la troisieme classe. En supposant 7 = 0, S = 0, ce qui reduit la courbe du troisieme ordre aux lignes i? = 0, S = 0, (S- a.) 8- SR — aS— 0, l'equation (15) se partage dans les deux equations 0 = 0,

A (G-B)a + B(C + A)G~0

(16),

et la courbe de la troisieme classe se re"duit au point (B, = 0, 8 = 0) et a une conique. Cela est un the"oreme connu, car Ton sait bien que si A, A' sont des points quelconques sur deux droites donnees a, a', et B, B' deux autres points de'termine's, sur ces droites, de maniere que l'intersection des lignes AB', A'B soit toujours sur une ligne droite donne'e, les deux lignes a, a' sont divisees homographiquement, B, B! e'tant deux points correspondants; et de plus, que la ligne qui unit les points correspondants de deux lignes divise'es homographiquement, est toujours tangente a une certaine conique. Quant au point d'intersection des lignes a, a', que nous venons de trouver comme formant avec la conique une courbe de la troisieme classe, on observera que, dans notre theorie, nonseulement les points des lignes a, a! se correspondent, mais que le point d'intersection des lignes a, a' correspond a tous les points de la troisieme droite. La conique touche les deux droites a, a'; cela n'a pas, je crois, d'analogie dans la the'orie ge'ne'rale.

190

[27

27. NOUVELLES EEMAEQUES SUR LES COURBES DU TROISIEME ORDRE. [From the Journal de MatMmatiques Pures et Appliquies (Liouville), torn. x. (1845), pp. 102—109.] J E me propose de deVelopper ici quelques consequences de la theorie que j'ai donnee, il y a quelques mois, dans ce Journal1, [26], des points correspondants des courbes du troisieme ordre. Rappelons d'abord la signification de ce terme et ajoutons-y quelques nouvelles definitions. On dit que les points A, A' sont correspondants quand les tangentes a la courbe en ces points se rencontrent sur la courbe. En consideYant les points correspondants A, A' et les deux autres points correspondants B, B', on dit que ces paires correspondent, quand les points d'intersection H, h de AB', A'B ou AB, A'B' sont situes sur la courbe. Les trois paires A, A', B, B', H, h sont nominees paires supplementaires ou systeme supplementaire. On dirait de meme que les deux systemes AA', BB', Hh et AiA'x, -Bi-B'i, HJii sont des systemes supple"mentaires correspondants, si, par exemple, A, A' et Alt A\, &c, formaient des paires correspondantes. On peut nommer quadrilatere inscrit le systeme de quatre droites qui passent par les points supplementaires AA', BB', Hh, et conique d'involution chaque conique tangente a ces quatre droites, ou, en d'autres termes, inscrite dans un quadrilatere inscrit. II est inutile d'expliquer ce que veulent dire coniques d'involution correspondantes, ou quadrilateres correspondants, ou l'expression conique correspondante a une paire donnee de points correspondants, &c. Demontrons les the'oremes suivants: TH^OREME I. " Les tangentes menees par un point P de la courbe a trois coniques d'involution correspondantes forment un faisceau en involution." 1

Voyez page 285 du tome ix.

27]

NOUVELLES REMARQUES SUE LES COURBES DU TROISIEME ORDRE.

191

Chaque conique peut se require a une paire de points qui correspondent aussi aux coniques (ce qui justifie la denomination que nous avons donnde a, ces coniques). Considerons un quadrilatere inscrit AA', BB', Hh, la conique d'involution tangente aux cotes de ce quadrilatere, et deux paires LL', MM' de points correspondants, ces paires etant correspondantes l'une a l'autre et a la conique d'involution. On sait que les lignes mene'es d'un point quelconque, et ainsi du point P de la courbe, par les points A, A', B, B', forment avec les tangentes a, la conique mene'es par ce me'me point, un faisceau en involution. Mais PA, PA', PB, PB', PL, PL', et de meme PB, PB', PL, PL', PM, PM', forment aussi des faisceaux en involution; done les deux tangentes forment, avec PL, PL' et PM, PM', un faisceau en involution. De me'me, avec une conique correspondante a la premiere, PL, PL', PM, PM' et les deux tangentes forment un faisceau en involution; done PL, PL' et les quatre tangentes forment un faisceau en involution, et de meme en introduisant la troisieme conique. THE'OREME II. " On peut circonscrire a une conique d'involution donne'e une infinite de quadrilateres inscrits correspondants au premier." Soient A, A', B, B', H, h' comme auparavant; et Ax, A\ une paire de points correspondants qui correspondent a ceux-ci; en menant par Alt A\ des tangentes a la conique qui se rencontrent en Blt B\, Hlt hu on voit d'abord que les lignes PA, PA', PA1} PA\ et les tangentes a, la conique forment un faisceau en involution; et reciproquement, chaque point P qui satisfait a, cette condition appartient a la courbe. Mais en prenant, par exemple, pour P le point B1} les lignes PAlt PA\ deviennent identiques avec les deux tangentes, ce qui satisfait a la condition d'involution. Done Blt B\, Hi, ^ appartiennent a la courbe, ou A1} A\, Blt B\, Hlt hx forment un quadrilatere inscrit correspondant au premier ou a la conique. TH^OREME III. "Les centres d'homologie de deux coniques d'involution correspondantes forment un quadrilatere inscrit correspondant aux coniques." ConsideYons les deux coniques et une troisieme conique quelconque. Chaque point P pour lequel les six tangentes forment un faisceau en involution appartient a la courbe. En prenant pour P un centre d'homologie des deux premieres coniques, les deux paires de tangentes deviennent identiques, ce qui satisfait a la condition d'involution ; done les six centres d'homologie sont sur la courbe, ou ces six points sont les sommets d'un quadrilatere inscrit qui correspond aussi aux coniques. Reciproquement, THE'OREME IV. " Le lieu d'un point P qui se meut de maniere que les tangentes mene'es par ce point a trois coniques donndes quelconques, forment toujours un faisceau en involution, est une courbe du troisieme ordre qui passe par les dix-huit centres d'homologie des coniques prises deux a deux, ces centres formant six a six des quadrilateres inscrits correspondants." La demonstration analytique de la premiere partie de ce the'oreme, quoique longue, me parait assez inte"ressante pour trouver place ici. Pour plus de symetrie, prenons t

v

%., i. pour les coordonne'es inddfinies d'un point, et representons par T=0,

T' = 0,

T" = 0,

192

NOUVELLES REMAEQUES STTR LES COTJRBES DU TEOISIEME ORDRE.

[27

les Equations des trois coniques, T, &c. etant des fonctions homogenes de la forme

T = Ap + By2 + C? + 2FVS+ 20ft + 2H£q, &c. Soient - , - les coordonnees du point P; mettons, pour abreger, z z IT = Ax2 + By* + Gz2 + 2Fyz + 2Gzx + 2Hxy,

(de maniere que W = 0 serait l'dquation de la ligne polaire du point P). Nous avons

ur - W2 = o, pour l'equation des deux tangentes mendes par le point P a la conique. Cela est probablement connu. II est clair d'abord que cette Equation appartient a une conique qui a un double contact avec la conique donnde, et par la forme a, laquelle nous allons re"duire cette Equation, on voit ensuite qu'elle appartient a un systeme de deux droites qui passent par le point P. En ddveloppant et e"crivant BG-

F> = a, GA-

GH-AF=f,

G* = b, AB-

HF-BG

H* = c,

= g, FG-CH = h,

on trouve, en effet, a (y{ - znf + b(z£- x%J + c(xrj- y%f + 2 / O | - xO (xn - yZ) + 2g (xv - y%) (y^-zr,) + 2h ( y ? - zv) (*f - x£) = 0, et de la, en posant & = bz* + cy3 - 2fyz,

§ = ayz- gxy - hxz +fx2,

23 = ex2 + az2 — 2gxz,

<Si = bzx — hyz — fxy + gy2,

(& = ay'i + bx* — 2hxy,

| ^ = cxy —fxz — gyz + hz2,

on obtient

a p + 2 3 ^ + c r - ajfac - 2ffif c - 2ffi%v = o, ou, en transportant l'origine au point P ,

et de rngme, pour l'equation des tangentes, par ce meme point aux deux autres coniques

On obtient done, pour la condition que ces lignes forment un faisceau en involution,

a , 23, a', 23',

.a", 23",

= 0,

27]

NOUVELLES BEMAEQUES SUE LES COUBBES DTJ TBOISIEME ORDEE.

193

en reprdsentant de cette maniere le determinant forme avec ces neuf quantites. Formons d'abord la fonction

Cela se reduit a z[-2 (fc) x2y- 2 (eg) xy2 + (ca)tfz- 2 (fa)yz2 - 2 (bg) xz* + (be)x2z + 4 (fg)xyz + (ba) z% o^ (fc) = (fc"-f"c'),

.fee.

Multipliant par T^, formant ensuite les quantites analogues et ajoutant; e'erivant aussi e'est-a-dire (cfg) pour le determinant forme avec les neuf quantit^s c,f,g, c',f,g', c",f",g", on obtient d'abord les termes \

+2(fcg)xyz\

+ 2 (gfc) a?fz\

qui se ddtruisent; les autres termes contiennent z3 comme facteur, et en ecartant cette quantity, Ton obtient en derniere analyse l'equation (cbf) a? + (acg) f + (bah) z* + 4< (fgh) xyz + [2 (agf) + (cah)] fz + [2 (bhg) + (abf)]z>x + [2 (cfh) + (beg)] x°y + [2 (afh ) + (abg)] yz* + [2 (bgf) + (bch)] za? + [2 (chg) + (ca/)] xf = 0 (ou Ton peut, si Ton veut, dcrire z = 1). C'est liquation d'une courbe du troisierae ordre. OBSERVATION. Cette methode peut etre utile dans l'investigation d'autres problemes relatifs aux coniques. Par exemple, la question de determiner les centres d'homologie de deux coniques donnees revient a celle-ci: satisfaire identiquement a l'equation

'Zv) = 0, parce que, en effectuant cela, il est facile de voir que - , - sont les coordonnees du centre cherche. Ecrivons a + ka' = a, c + kb' = b, &c, l'on obtient les six equations bz2 + cy2 — 2fyz = 0, ex2 + &z2 — 2gzx •= 0, ay2 + ha? — 2hxy = 0, C.

&yz — gxy — hxz + ix2 = 0, bzx — hyz — iyx + gy2 = 0, cxy — izx — gzy + hz2 = 0, 25

194

NOUVELLES REMARQUES SUR LES COURBES DU TROISIEME ORDRE.

[27

dont les trois dernieres se de'duisent des autres. On peut, de ces six Equations, eliminer les six quantity's a?, y2, z2, yz, zx, xy, conside're'es comme independantes; on obtient ainsi, toute reduction faite, (abc -at2-bg2-

ch2 + 2fgh)2 = 0,

equation qui determine la quantity k; les trois premieres equations deviennent alors equivalentes a deux, qui suffisent pour determiner les rapports - , —. II serait facile Z

2

de rapprocher cette solution de celle que l'on ddduit de la theorie des polaires reciproques. Par exemple, liquation qui vient d'etre obtenue entre les quantites a, b,... est precisement celle qui exprime que la fonction ax2 + by2 + cz% + 2iyz + 2gxz + 2hsuy

se divise en facteurs lineaires. Remarquons encore que, dans la ge'ome'trie solide, liquation

appartient au c6ne, ayant le point P pour sommet, et circonscrit a une surface du second ordre qui a pour Equation T = 0. C'est un cas particulier d'une autre formule que voici: " En repr^sentant par ^ une fonction line'aire des trois variables £, t], % (ou de quatre variables sans termes constants), et par P la meme fonction de x, y, z, l'equation

appartient au c6ne ayant pour sommet le point dont les coordonnees sont x, y, z, et passant par la courbe d'intersection du plan P = 0, et de la surface du second ordre T = 0. Et de meme pour deux variables." Je finirai en citant un th^oreme de ge'ome'trie du a, M. Hesse1, qui a quelques rapports avec le sujet que je viens de traiter: " Le lieu d'un point P, qui se meut de maniere que ses polaires par rapport a, trois coniques donnees se rencontrent dans le meme point, est une courbe du troisieme ordre; et encore cette courbe ne change pas quand on remplace les coniques donne'es U=0,

U'=Q,

U" = 0,

par trois nouvelles coniques de la forme \U+\'V + \"U" = 0" Cette courbe du troisieme ordre est tout a, fait distincte de celle que j'ai consid^ree. 1

Voyez le Journal de M. Crelle, tome xxvm.

28]

195

28. SUR QUELQUES INTEGRALES MULTIPLES. [From the Journal de Mathematiques Pures et Appliquees (Liouville), tome x. (1845), pp. 158—168.] J'AI donnd, il y a trois ans, dans le Cambridge Mathematical Journal, [2], une formule assez singuliere pour l'integrale multiple

prise entre les limites donne"es par l'equation x2 7 n

a;2 I

7 n ~T • • •

J-J

la fonetion $ e"tant seulement assujettie a la condition de ne pas devenir infinie entre les limites de l'inte'gration. L'expression que j'obtiens est une suite infinie, dont le terme gdndral est de cette forme,

En appliquant ce re"sultat a un cas particulier, j'ai obtenu l'inte'grale a n variables analogue a celle qui exprime le potentiel d'un ellipsoide homogene, pour un point exte"rieur. J'ai depuis cherche" a etendre ces rdsultats au cas d'une density variable et e'gale a, une fonetion rationnelle et entiere des coordonnees, et d'une loi d'attraction selon une puissance quelconque de la distance (toujours a. n variables); mais quoique j'aie re"ussi a effectuer cette generalisation, mes formules etaient si confuses et si infe'rieures a, celles que donne la belle analyse de M. Lejeune-Dirichlet, que je ne les ai jamais publiees. Cependant, en revenant il y a quelques jours sur ce sujet, en

25—2

196

SUR QUELQUES INTEGRALES MULTIPLES.

[28

me fondant sur une integrate plus ge"ne"rale, j'ai trouv^ que la question e'tait a peine plus difficile que dans le cas d'une densite" constante, et se laissait traiter exactement de la m&me maniere. J'ai reussi de cette facon a. exprimer Finte'grale' cherchee au moyen d'une seule integrate definie abelienne, et de ses coefficients differentiels relatifs aux constantes qui y entrent; et il m'a para que les formules que j'ai ainsi obtenues pourraient n'etre pas tout a fait indignes de l'attention des ge"ometres. Conside"rons l'inte'grale multiple a n variables V, donnde par liquation V = I drj ... a;12ai+1 ... ( / termes) */+i2o/+i... (n —/ termes) (e^ — xxt,...), oil les variables xlt ... doivent recevoir des valeurs re"elles quelconques, positives ou negatives, qui satisfassent a la condition —+

<1-

et Ton suppose de plus qu'il est permis de d^velopper (sous le signe integral) la fonction suivant les puissances ascendantes des- variables a^,.... En faisant ce d^veloppement, il est clair que les termes qui contiennent des puissances impaires d'une ou de plusieurs des variables se ddtruisent par l'inte'gration; done, en ne faisant attention qu'aux termes qui contiennent seulement des puissances paires, on a ce terme ge"ne"ral,

[2r/+J-w ... [daj

''' \dafj

-

rr

x I dxz. ou

p = n + ... ;

il est a, peine necessaire de remarquer que, dans l'expression

il faut prendre / termes tels que \%rx +1]^'4"1, et n — f [2r/+1]2r/+i, et ainsi dans tous les cas semblables.

termes de l'autre forme

L'inte'grale qui entre dans cette formule a etd trouvee, comme on sait, par M. Dirichlet. Sa valeur est

r K + n + § ) • • • r («

+s

* ou.

28]

SUB, QUELQUES INTEGRALES MULTIPLES.

197

Le terme entier devient, apres quelques reductions tres-simples,

(p+k +/+ \n

ou Ton ecrit, pour un moment,

Z,

=(2r, +3) (2n + 5 ) . . . (2r, +2a, +1)

Mf+1 = (2r/+1 +1) (2r /+1 + 3) ... (2r/+1 + 2a/+1 puis on le transforme en +f T(p+k X

-af,

En effet, les symboles -^— et h^^j- dtant conversibles, on a J da dht

et ainsi de suite. En prenant la somme de tous les termes qui correspondent a une meme valeur de p, on a

En posant

puis, en observant que p doit s'dtendre depuis 0 jusqu'a oo, on a, pour l'integrale cherche'e,

2k+f Kda/'-daJv1

dhj

-

n + 1) ce qui fait voir que l'integrale V depend de cette seule expression,

198

SUB QUELQUES INTEGRALES MULTIPLES.

[28

On peut remarquer, en passant, que cette quantity satisfait a cette Equation differentielle,

M. G. Boole, de Lincoln, a ddduit une equation semblable de mes formules dans le Mathematical Journal. O'est a. lui qu'on doit l'introduction dans l'integrale proposed de cette quantite t, ce qui, au reste, n'est pas d'une grande importance ici; mais j'ai cru devoir la conserver, a, cause de cette Equation me'me, qui pourrait conduire a des resultats interessants. A present, soit

ou s est un nombre entier; je pose, pour abrdger, fan-— s = i, h + / + s = a; ce qui donne

et ainsi

• i + of Le cas de a = 0, qui est le plus simple, est, en effet, celui du Me"moire cite", et a ce cas on peut rdduire celui de cr entier negatif; il y a me'me deux manieres d'effectuer cette reduction. Representons, en effet, la valeur de U par cette notation plus complete Ui; on obtient tout de suite 1

et

\t

UC = 2,
(£*+~)'

U

<-">

la seconde desquelles equations se deduit de cette formule facile a demontrer,

Nous pouvons done, dans la suite, ne conside"rer que le cas de a entier positif. Commencons par ope'rer la transformation que voici; en determinant f par l'equation

28]

SUR QUELQUES INTEGRALES MULTIPLES. h = ~z. > •••}

je pose

a

199

i= (1•

ce qui donne at+... = V[{\ + k)V + ...\ t*\l+lj

day-")'

ou il faut remarquer que ce £, contenu en V, ne doit pas etre affecte des symboles -=- . . . . de V, de maniere 1qu'il faut ^crire dax

\2P[p + i + af+'+i [p]PKl

''

\\ + l1da?'r---)

{ai2(l + k) + ..

formules qui se present mieux aux reductions, quoique plus complique'es en apparence. Ecrivons d'abord

En developpant la pKlBe puissance de ce symbole selon les puissances de A, on a

qui doit s'appliquer a Considerons l'expression

et mettons, pour un moment, cette expression se trouve reduite a. ,+ ... laquelle, par une formule deja, citde, devient +p-q-

ljM. [i + | ) - ? - ^

] P

-g__i___

200

SUE QUELQUES INTEGRALES MULTIPLES.

c'est-a-dire

<W~n [i+p-q-

1J-9 [i + p-q-

[28

fr

Le terme general de U, en faisant abstraction du facteur

^

8f

^ , se reduit a

a

considerons la partie de ce terme qui est de l'ordre 6 en llt ... , elle devient

Soit, en general, © une fonction homogene de l'ordre 20 des lettres alt...;

on a

et de la, en repetant toujours l'opdration A, et faisant attention a, ce que A©, A2 sont des fonctions homogenes des ordres 20—1, 20 — 2, &c, on obtient

[i + q-20-

in

depuis X = 0 jusqu'a X = q. Puis, pour le terme general de U, ^

'

2* [p - q¥>-i [0f [X]A [q - X]i-K

r,' j . n L F

i a

TL

i "ie—a—A—i

J

1

28]

ST7R QUELQUES INTEftBALES MULTIPLES.

201

le dernier facteur etant independant de q, q doit s'e'tendre depuis 0 jusqu'a p. Mais a cause de [q — X]*~x qui devient infini pour q<\, on peut faire etendre q depuis q = A. jusqu'a q=p ', ou, en e'crivant q-\ = q', p-\=p't q' s'etend depuis 0 jusqu'a p'. Le facteur a sommer est done

si, pour un moment, Ton pose C=i + p'-\n, Cette somme se reduit a.

e'est-a-dire a

et nous avons ainsi le terme general _ X]e-P {i + e - X ^

J

Ecrivons p = 6-p'; cela devient g-j,1]^

ou

(-)•

M=6-\, C = i+20-\-l,

C-A=6-a-X-l;

p' doit s'etendre depuis — oo jusqu'a 9. Mais a, cause de [p'Y> qui devient infini, pour p' ndgatif, on peut l'etendre seulement depuis 0 jusqu'a 6, ou meme seulement depuis 0 jusqu'a M, a cause du facteur [ilf]*'. La somme se re'duit a [C - M]C-M-A rfl _ Ayt

=

p + Q _ !]-„-, ^ _ a _ x _ !],_*

et le terme general devient

Ecrivons enfin d=\+K, c

-

26

202

SUR QUELQUES INTEGRALES MULTIPLES.

[28

le terme devient

II faut que K soit toujours positif, car autrement le terme s'evanouit a cause de AA(Z1a12 + •••)K+K; mais pour K plus grand que a, le facteur [K - a — 1]" s'evanouit, done K s'etend seulement depuis 0 jusqu'a
ou

g2 + ... = X,

c'est-a-dire

q1 + z^J '... . LW-L'/x-W--"-* ou, en re'duisant, 22X [Xf et cela donne pour V,

[<

(-Y Soit

on a (X) = I et de la, 1

1

de maniere que le terme de U devient

y . . . I 1 dlj ''

:

V[» + * + X

Prenant la somme pour X, a l'aide de

(ce qui suppose I + u > 0), on obtient

k

.

k

-

0 1

'•

qui contiennent afk>,...;

28]

SUE, QUELQUES INTEGRALES MULTIPLES. 22" [a]'

[K-
1]* «!*...

1

/ 1

2

d \*'

203

f1 (1 - uf u1*'-1 du

" . . . ( « , • + . . . y ~ 71^7.V dTJ • • • ^ • • • J h2 ou, mettant lx — ~ ,... ,

h1...[hldhj en r^tablissant le facteur constant

r

...

2j

-•hl-J

.^ de f7, le terme general de cette quantite est

^ r (o ou fei, &c, sont des entiers positifs quelconques qui satisfont a k, + ... =*, et K peut s'dtendre depuis 0 jusqu'a a. II faut observer qu'en diffeVentiant par -yj-, &c, on ne doit pas conside'rer ^ comme fonction de hx...,

ce qui est cependant

ndcessaire dans liquation entre V et U.

26—2

204

[29

29. ADDITION A LA NOTE SUR QUELQUES INTEGRALES MULTIPLES. [From the Journal de Mathematiques Pures et Appliquees (Liouville), tome x. (1845), pp. 242—244.] ON ddmontre, au moyen des formules que j'ai donnees sur ce sujet, [28], une propriete remarquable de l'inte'grale multiple V=

I dxx ... dXnOO^1

... «/ + i 2a /+i . . . < / > ( « ! -

Xjt,...),

prise entre les limites

En effet, en supposant toujours que Ton peut deVelopper la fonction
V= SP~o (2* r (p

Soit

d \»'

29]

ADDITION A LA NOTE SUE. QUELQUES INTEGRALES MULTIPLES.

205

entre les memes limites que V. On a

En multipliant par

et integrant depuis 7 = 0 jusqu'a 7 = 1 , on obtient 1

{

{l

>

(p l)T(p + k+f+ \n + 1 ) V ^ ^° 1 ''' 'V

puisque en ge'ne'ral

1 f V(k+f) 2

On a done l'equation

r(&+/) Mettons la valeur de U qui en re'sulte, dans l'equation entre F et U; faisons aussi t = l, ce qui ne nuit pas a la ge'neralite'. On a, en rassemblant les formules, r

W = jdx1...dxn...

^(oj-a^Ti...),

ce qui dtablit ce theoreme: La suite des integrates V s'exprime au moyen des coefficients diffeYentiels par rapport a, h^ ...hn et Oi... a/ de la seule integrale x,...

dxn ... (j>(a,-

x,T,...).

En supposant que les indices dans V soient tous pairs, on a / = 0.

Les symboles

-j—, ••• n'entrent plus dans l'expression de V. En changeant la fonction , on a ces formules plus simples,

206

ADDITION A LA NOTE SUE, QUELQUES INTEGRALES MULTIPLES. V =

dx, ... dxnxi*i

... xn2a" 4> ( » ! , . . . ,

W=jdx1...dxn4>(Txu...,

v

xn),

Txn),

• • • [^ i X [

=

[29

T

n+1

^ -

T

^

WdT

>

ou V s'exprime au moyen des coefficients diffe'rentiels par rapport a hu..., hn de l'int^grale

f

Tx,,...,

Txn),

de maniere que nous avons trouve" des relations entre des inte'grales ddfinies qui contiennent une fonction inddterminee, assujettie a la seule condition d'etre deVeloppable dans les limites de 1'integration selon les puissances entieres et positives des variables1, mais dans lesquelles les limites sont donnees par ,X3

1 Cette condition est supposee dans la demonstration, mais il me paralt, d'apres la forme da resultat, qu'elle n'est pas essentielle.

30]

207

30. MEMOIRE SUR LES COURBES A DOUBLE COURBURE ET LES SURFACES DEVELOPPABLES. [From the Journal de Mathematiques Pures et Appliquees (Liouville), tome x. (1845), pp. 245—250. ON trouve, dans la Theorie des courbes algebriques de M. Pliicker, de tres-belles recherches sur le nombre des diffe'rentes singularity's (points d'inflexion, tangentes doubles, &c.) des courbes planes. Les memes principes peuvent s'appliquer au cas des courbes a trois dimensions. Pour cela, considerons une suite continue de points dans l'espace, les lignes qui passent par deux points consecutifs, et les plans qui passent par trois points conse'cutifs; ou, en envisageant autrement la meme figure, une suite de lignes dont chacune rencontre la ligne consecutive, les points d'intersection de deux lignes conse'cutives, et les plans qui contiennent ces lignes; ou encore de cette maniere : une suite de plans, les lignes d'intersection de deux plans consecutifs, les points d'intersection de trois plans conse'cutifs. C'est ce que l'on peut nommer un systeme simple. Ce systeme est eVidemment forme d'une courbe a double courbure et d'une surface developpable; la courbe est l'arete de rebroussement de la surface, la surface est l'osculatrice developpable de la courbe. Les points du systeme sont des points dans la courbe, les lignes du systeme sont les tangentes a la courbe, les plans du systeme sont les plans osculateurs de la courbe. De meme, les plans sont les plans tangents de la surface, les lignes sont les generatrices de la surface ; pour les points, on peut les nommer les points de rebroussement de la surface. J'entendrai dans la suite par les termes ligne par deux points, ligne dans deux plans, une ligne menee par deux points quelconques (non conse'cutifs en general) du systeme, et la ligne d'intersection de deux plans quelconques (non conse'cutifs en general) du systeme. Enfin, chaque ligne du systeme est rencontree, en general, par un certain nombre d'autres lignes non consecutives du systeme. Je nommerai le point de rencontre d'une telle paire de lignes point dans deux lignes, et le plan qui contient une telle paire plan par deux lignes.

208

MEMOIRE SUE, LES COURBES A DOUBLE COURBURE

[30

Supposons qu'un plan donne contient, en general, m points du systeme, qu'une ligne donnee rencontre, en general, r lignes du systeme, qu'un point donne est situe', en general, dans n plans du systeme. Le systeme est dit e"tre de l'ordre m, du rang r, de la classe n. On voit tout de suite que l'ordre de la courbe est egal a l'ordre du systeme, ou a TO; la classe de la courbe au rang du systeme, ou a r. Et de meme, l'ordre de la surface au rang du systeme, ou a r; la classe de la surface a la classe du systeme, ou a n. Cela pose, les singularity's proprement dites (ouvrage cite", page 202) sont les deux suivantes, analogues aux points d'inflexion et de rebroussement dans les courbes planes: 1. Quand quatre points consecutifs sont situds dans le meme plan, ou, autrement dit, quand trois lignes conse'cutives sont situees dans le me'me plan, ou quand deux plans consecutifs deviennent identiques; je dirai qu'il y a alors un plan stationnaire, et je representerai par la lettre a le nombre de ces plans. 2. Quand quatre plans consecutifs se rencontrent dans le meme point, ou, autrement dit, quand trois lignes se rencontrent au m6me point, ou quand deux points consecutifs deviennent identiques; je dirai qu'il y a alors un point stationnaire, et je representerai par § le nombre de ces points. Dans le premier cas, il y a un point d'inflexion sphe'rique dans la courbe, et une ligne d'inflexion dans la surface. On n'a pas donne" de noms a, ce qui arrive dans la courbe et la surface dans le second cas; et puisque la singularity est suffisamment distingue'e deja en la nommant point stationnaire, il n'est pas necessaire de suppleer a cette omission. On peut dire que ces deux cas sont les singularites simples d'un systeme. II y a des singularites d'un ordre plus eleve dont on n'a pas besoin ici. Ensuite, il y a des singularity d'une autre espece, en quelque sorte analogues aux points et tangentes doubles, mais qui ont rapport a un point ou plan indetermine' (hors du systeme) ; savoir : 3.

Un plan donne* peut contenir, en general, un nombre g de lignes dans deux plans.

4. Un point donne* peut etre situe, en general, dans un nombre h de lignes par deux points. 5. Un plan donne peut contenir, en general, un nombre x de points dans deux lignes. 6. Un point donne peut 6tre situe', en general, dans un nombre y de plans par deux lignes. Ces quatre cas sont les singularites impropres simples du systeme. II faut maintenant chercher les relations qui ont lieu entre les nombres TO, r, n, a, S, g, h, x, y. Citons d'abord les formules de M. Pliicker pour les courbes planes (page 211), en changeant seulement les lettres, pour eViter la confusion. En representant par p l'ordre d'une courbe, par v sa classe, par £ le nombre de ses points doubles, rj de ses points

30]

ET LES SURFACES DEVELOPPABLES.

209

de rebroussement, a de ses tangentes doubles, b de ses points d'inflexion, Ton aura les six equations

-|

v

/O/-# _j_ QZA

= Sv. v - 2 - (6a + 86),

% = % v .v-

2 v3 - 9 - (2a + 3b) (v. v - 1 - 6) + 2a .a -I

+ %b .b -I + 6ab,

dont les trois dernieres se derivent des trois premieres, et vice versa. Considerons ensuite un plan donne quelconque en conjonction avec le systeme. Ce plan coupe la surface suivant une courbe plane. Les points de cette courbe sont les points de rencontre du plan avec les lignes du systeme, les tangentes de cette courbe sont les lignes de rencontre du plan avec les plans du systeme. II est clair que la courbe est de l'ordre r et de la classe n. Chaque fois que le plan contient un point en deux lignes, la courbe a un point double; aux points ou le plan rencontre la courbe a double courbure, il y a dans la courbe un rebroussement (car, dans ce cas, il y a deux lignes du systeme qui coupent le plan en un meme point, c'est-a-dire qu'il y a dans la courbe d'intersection un point stationnaire ou de rebroussement). Quand le plan contient une ligne en deux plans, il y a dans la courbe une tangente double; et enfin, pour chaque plan stationnaire du systeme, il y a dans la courbe une tangente stationnaire ou une inflexion. Done nous avons, dans la courbe, r l'ordre, n la classe, x le nombre de points doubles, m de points de rebroussement, g de tangentes doubles, a de points d'inflexion, ce qui donne les six equations (equivalentes a trois): n = r . r — 1 — (2x + 3m),

a = 3r. r - 2 - {fax + 8m), — 2 r 2 - 9-(2x + 3m)(r.r— 1 —6) + 2 « . a r - l + | m . m - l + Qxm; = n.n—l — (2g + 3a), = Sn.n-2-(Qg

+ 8a), 2

x = \ n. n - 2 n - 9 - (2g + 3a) (n. n - 1 - 6) + 2g . g - 1 + 1 a . a - 1 + 6gu (ou Ton peut remarquer la syme'trie a l'egard des combinaisons n, a, g, et r, m, x). De meme, en conside'rant un point donne quelconque en conjonction avec le systeme, ce point determine, avec la courbe a double courbure, une surface conique, et, par des raisonnements pareils, on fait voir que, dans cette surface conique, l'ordre est m,

c.

27

210

MEMOIRE SUE LES COURBES A DOUBLE COUftBURE

[30

la classe r, le nombre des lignes doubles n, des lignes de rebroussement 8, des plans tangents doubles y, des lignes d'inflexion h, ce qui donne les Equations (dont trois seulement sont inde"pendantes):

n = 3m.m-2-(6h + 88), y = I m. m - 2 m2 - 9 - (2h + 38) (m. m - 1 - 6) + 2h. h - 1 + § §. § - 1 + 6h8; wi = r . r -~ 1 - {2y + 3»),

A. = I r. r - 2 r2 - 9 - (2y + Bn) (r. r -1 - 6) + 2y. y -1 + f n. n -1 + 6yn (dans lesquelles on remarque la correspondance r, n, y; m, §, h: et, en les comparant avec les autres six equations, la correspondance m, r, n, a, S, g, h, x, y; n, r, m, §, a, h, 9,

V, «>)•

En considerant une courbe a double courbure d'un ordre donne" m, on peut attribuer k h, § des valeurs quelconques (entre certaines limites), et Ton a alors, pour determiner les autres quantitfe, les Equations suivantes: r = m . m - l - (2h + 38), n=3m. m - 2 - (Qh + 88),

n = r. r — 1 — (2x + 3m), a = 3r . r - 2 - (6* + 8m), g = \r .r-2r2

— d- (2x + 3m) (r.r-l-6)

+ 2x.x-l+§m.m,-l

+ <6xm.

Au cas d'une courbe plane les trois premieres equations continuent d'etre vraies, mais les trois autres n'ont pas de sens. Le cas le plus simple est celui d'une courbe du troisieme ordre dans l'espace. On a, dans ce cas, m = 3, n = 1, § = 0; et de la- le systeme in, n, 3,

r,

a, £, g, h, x,

3, 4, 0,

0,

1,

1, 0,

y; 0;

c'est-a-dire l'osculatrice deVeloppable d'une courbe du troisieme ordre est une surface du quatrieme ordre, &c. On obtient aussi un systeme tres-simple, en e"crivant m = 4, h = 2, € = 1, ce qui donne m,

n,

r,

a, 8, g,

4,

4,

5, 1,

1,

2,

h, x, 2,

y;

2, 2.

30]

ET LES SURFACES DEVELOPPABLES.

211

Mais cela n'appartierit pas, a ce que je crois, aux courbes les plus ge'ne'rales du quatrieme ordre. Le probleme de classifier les courbes a double courbure au moyen des surfaces que Ton peut faire passer par ces courbes, ou de trouver la nature d'une courbe qui est l'intersection de deux surfaces donne'es, paralt appartenir plutdt a la the'orie des surfaces qu'a celle des courbes. Je n'ai rien de complet a offrir sur cela. Seulement je crois pouvoir dire que quand la courbe d'intersection de deux surfaces des ordres /i, v est de l'ordre fiv (ce qui est le cas general), on a toujours %h = }iv. [i — 1 v — 1,

de maniere que la classe de la courbe est au plus mn(m+n— 2). Mais j'espere revenir une autre fois sur cette question. II est presque inutile de remarquer que pour un systeme qui est re"ciproque polaire d'un systeme donne, il faut seulement changer m, r, n, a, 8, g, h, x, y en n, r, m, §, a, h, g, y, x. Par exemple, les deux systemes qui viennent d'etre conside're's ont des re'ciproques de la meme forme.

27—2

212

[31

31. DEMONSTRATION D'UN THEOKEME DE M. CHASLES. [From the Journal de Mathematiques Pures et Appliquees (Liouville), tome x. (1845), pp. 383—384.] " Soient P, P' des points correspondants de deux figures homographiques; si la droite PP' passe toujours par un point fixe 0, les points P sont situes sur une courbe du troisieme degre, qui passe par ce mfime point." Soient — , — , — les coordonne'es de P; —,, —., —, celles de P'. En supposant rr w w w w w w que x', y', a!', w' sont des fonctions lineaires (sans terme constant) de x, y, z, w, les deux figures seront homographiques. Soient, de meme* ^, ^, % les coordonne'es de 0; — , —, — les coordonne'es d'un O

O

'ST

O

W

UT

point quelconque T. Puisque P, P', 0 sont sur la meme droite, on peut faire passer un plan par les quatre points P, P', 0, T. Cela donne tout de suite liquation fl,

V,

a,

G,


x,

y,

z,

(X,

y,

y', /,

IS

w

w' J

(en representant de cette maniere le determinant forme avec les quantites X, fi, v, &c), Equation qui doit 6tre satisfaite quels que soient \, /x, v, ts, et qui equivaut ainsi aux deux conditions

(a,

G, 7 ]

f a , G, S ^

< x,

y,

< x,

z >=0,

U, y', A

y,

w Y=0.

U', y', vf)

Ces deux dernieres equations sont du second degre par rapport aux quantites - , —, - . Le point P est done situe a l'intersection de deux surfaces du second ordre. Mais ces surfaces ont en commun la droite repre'sente'e par les equations ay-gx= 0, ax' - §y' = 0. Done elles se coupent de plus suivant une courbe du troisieme degre qui passe evidemmenfc par le point 0, parce qu'on satisfait aux equations en ecrivant x : a = y : § = z : y= w : S.

32]

213

32. ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION TO THE THEORY OF SPHERICAL COORDINATES. [From the Cambridge and Dublin Mathematical Journal, vol. i. (1846), pp. 22—33.] SECTION 1.

THE formulae in question are only very particular cases of some relating to the theory of the transformation of functions of the second order, which will be given in a following paper. But the case of three variables, here as elsewhere, admits of a symmetrical notation so much simpler than in any other case (on the principle that with three quantities a, b, c, functions of b, c; of c, a; and of a, b, may symmetrically be denoted by A, B, C, which is not possible with a greater number of variables) that it will be convenient to employ here a notation entirely different from that made use of in the general case, and by means of which the results will be exhibited in a more compact form. There is no difficulty in verifying by actual multiplication, any of the equations here obtained. It will be expedient to employ the abbreviation of making a single letter stand for a system of quantities. Thus for instance, if 8 = 6, (j>, yjr, this merely means that * (8) is to stand for <£ (6, , yjr), kS for kO, kcf>, kyfr, &c.

Suppose then ® = £> v, ?>

(i)»

Q = A, B, C, F, G, H

(2),

W (», »', Q) = A& + Bvv' + C& + F W + V'O + G (Cf + rf) + H (&' + ?»;) ... (3), the function W satisfies a remarkable equation, as follows:

214

ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION

write

&=BC-F\ 33 = CA

[32 (4),

-G\

1%=FG -CH.

® = a, as,
(5),

w=?rwr, vs-n, w-?v

(6),

we have W(alt

a>2, Q)W(w3,

» 4 , Q)- W (a*,, u t , Q) Tf(«j, »«, Q ) = T T ( ^ , «2«3, © ) ... (7);

of which we may notice also the particular cases ,
Q)3> Q)=W(^ws,

^ 3 , @) ... (8),

W(OH, »i, Q)TT(oHi, «2) Q ) - { F ( » 1 , «2, Q ) } 2 - T f ( ^ 2 , ^ , ffi) ... (9). To these we may join the following formulae, for the transformation of the function W. Suppose % + b"z1,

then, writing

g =a , b , 9'=B,',

c^ + c'yx + c"z1

c

(10),

(11),

b',

c',

g" = B,", b",

c",

Pi = Xi, Vi, Z\

(12),

p2^os2, y2, z2, ®=W(g,g,Q),

W(g',g',Q),

W(g",g",Q),

W(g',g",Q),

W(g",g,Q),

W(g,g',Q) (13),

we have , «2) Q)=W(Pl,

p,, ©)

(14).

Similarly, writing

^

we have

W(i^,

,
~<^, <&)=W(p}pA, p#h, W)

in which equations © may obviously be changed into Q.

(15),

(16),

TO THE THEORY OB1 SPHERICAL COORDINATES.

32] SECTION

215

2. Geometrical Applications.

Consider any three axes Ax, Ay, Az, and let X, /A, V be the cosines of the inclinations of these lines to each other. Let A, M, N be the inclinations of the coordinate planes to each other; I, m, n the inclination of the axes to the coordinate planes. Suppose, besides, a = l -X 2

(17),

2

t = l -/x , t = 1 - v\ f =/MV-\,

k=l

- Xs - ^ a - i/5 + 2\/u/

(18);

we have the following systems of equations: V(t)C)cosA=-f,

V(bt) sin A = V(k),

V (ca) cos M = - g, V (ca) sin M = A/ (k), V (at)) cos N = - f),

V (a)sin I = V(k)

(19).

V (fo) sin m = V (k)

V (aft) sin N = V (A), V (C) sin w = V (k). a + vi) + fi% = k, VB.

(20).

+ ft + xg = 0,

fiB. + Xfi +

8 = 0.

+ Atf=O,

/u.fi+Xft+

f=0.

g+ wf+ ^ = 0, v%+ f+ XC=0. fia + xf + ftt-

(21).

(22).

c = &. i* = ka

c a - f = M>, a b - ft2 = A;c, gfi - af = ifc/, ftf - ftfi = kg, fg — eft = kh, fc2 (24).

(23).

21G

ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION

[32

Imagine now a line AO, and let a, /3, 7 be the cosines of its inclinations to the three axes. Suppose also, 8, j>, x being its inclinations to the coordinate planes, we write sin 6 a = -rr^-,

sinrf) 0=—TTEV,

7

V(a)'

siny c= -r7^

V(b)

V(c)

._.. (2o). v

'

If we consider a point P on the line AO, at a distance unity from the origin, we see immediately, by considering the projections in the directions perpendicular to the coordinate planes, that the coordinates of this point are a, b, c. By projecting on the three axes and on the line AO, we then obtain the equations a = a + vb + fie

(26),

/8 = va + b +\c, 7 = fia + Xb + c, 1 =aa+/36 + yc

(27),

from which we obtain (28), kc = ga + f/3 + £7, l = aa+/3b + yc

(29),

and hence 1 = a* + ft + & + 2Xbo + 2fiac + 2vab

(30),

&=aa 2 +b/3 2 + C72 + 2f/37 4-2ga7 + 2l)a/3

(31).

Hence writing a, b, c=t

(32),

a, A 7 = ^

(33),

1, 1, 1, X, p , v =
(34),

a, b, t, f, g, 6 = q

(35),

we have the equations , t, <J)

(36),

k=W(r, r, q)

(37).

Let A0' be any other line, and <5 its inclination to AO: a.', /3', y', a', b', c, the quantities corresponding to a, /3, 7, a, b, c, and similarly t', T to t, T. We have of course 1 = W(t', H, q)

(38),

k=W(T', T', q)

(39).

32]

TO THE THEORY OF SPHERICAL COORDINATES.

217

We have besides, by projecting on the line AO', the equation cos8 = a'a+/3'b+
(40),

or the analogous one (41)From either of which we deduce cos 8 = aa' + bb' + cc' +X (be' + b'c) + fi (ca' + c'a) + v (ab' + a'b) k cos 8 = aaa' + b/3/3' + C77' + f (£7' +13'y) + 8 (7a' + 7V) + & (a/3 + a'/3)

(42), (43);

which may otherwise be written cosS=TT(i!, t', q) & C O S 8 = F ( T , T,

5)

(44), (45);

or again, observing the equations which connect the quantities t, T, * W(t, H, q) C0S ° \j{W(t, t, q) W(1f, t', q)}

forms which, though more complicated, have certain advantages; for instance, we derive immediately from them the new equations

, W,
wl¥777^] ,rf, q)} written more simply thus sinS=

W{t?_, W, 5)

(50),

^ ' , W, q)}

(51);

^U)

(53)

to these we may join

T , TT', SECTION

3.

q)}

On Spherical Coordinates.

Consider the points X, Y, Z, on the surface of a sphere, as the intersections of the three axes of the preceding section, with a sphere having its centre in the origin. It is evident that X, /A, V are the cosines of the sides of the spherical triangle XYZ, C. 28

218

ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION

[32

A, M, N are its sides, I, m, n are the perpendiculars from the angles upon the opposite sides. Let P be the point where the line A 0 intersects the sphere: the position of the point P may be determined by means of the ratios £ : 77 : f, supposing f, t], £ denote quantities proportional to the a, fi, 7 of the preceding section, i.e. f : V : £ = c o s P X : cos PY : cos PZ

(54);

or again, by means of the ratios x : y : z, supposing x. y, z denote quantities proportional to the a, b, c of the preceding section, i.e. sin Px sin P« sin Pz x : y : z =—.—^- : —=—£ : —.—w " sin X sin 7 sin Z

,_ _N (55), v '

(Px, Py, Pz are the perpendiculars from P on the sides of the spherical triangle XYZ). These last equations may be otherwise written, x sin X

sin PZY

. . ^ >'

ysmY smPXZ 'zwa.Z~wa.PXY' z sin Z_sin PYX x sin X sin PYZ ' The ratios f : rj : £ or x : # : z, are termed the spherical coordinate ratios of the point P . The two together may be termed conjoint systems: the first may be termed the cosine system, and the second the sine system. The coordinates of the two systems are evidently connected by £ : 77 : £=x +vy + fiz : vx + y +~kz : fix + \y+z

or

x : y : * = af +fo + tf •ftf+ fy+ff : 9? + in +C?

(57),

(58).

The systems may conveniently be represented by the single letters

« = £ v, K P = a, y, z

(59), (60).

Fundamental formula of spherical coordinates; distance of two points. Let P , P ' be the points, & their distance, w, p the conjoint coordinate systems of the first point, »', p' of the second; we have obviously '> q) , q) TT(p', / , q)}

V{F(g, g , q)} -J{W(p,p,q)W(p',p',q)}'

/-fin

{

h

32]

TO THE THEORY OF SPHERICAL COORDINATES. x coso = ,,-nr, . ',-,—-,—vr
or

219 (o2).

jg, jg, g)} V{Tf (», «, q) T F K »', q)}'

', W , q)} Equation of a great Circle. Let the conjoint coordinate systems of the pole be e = a, &, c

(63),

€ = 0,0, y

(64),

then, expressing that the distance of any point P in the locus from the pole is equal to 90°, we have immediately the equations W(p, e, q) = 0

(65),

F ( » , e, q) = 0

(66),

which may otherwise be written in the forms 0

(67),

0

(68),

or the equation of a great circle is linear in either coordinate system. any linear equation belongs to a great circle.

Conversely,

Suppose the equation given in the form Ag + Bfi + Cg=0

(69);

or by an equation between cosine coordinate ratios:—the sine system for the pole is given by e = A, B, C (70), and the cosine system by e = A+vB+fj.C,

vA + B + XG, pA + XB + C

(71).

Suppose the circle given by an equation between sine coordinates, or in the form y + Gz = 0

(72),

the cosine system of coordinates for the pole is given by e = A, B, C

(73),

and the sine system by (74). 28—2

220

ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION

[32

It is hardly necessary to observe, that if At; + Bri+CS=O Ax + By+Cz = 0

(75), (76),

represent the same great circle, A : B : C = A+vB + pC : vA+ B + XG : fiA+\B+

C

(77),

A : B : C = a A + |)B+gC : &A + bB + fC : gA + f B + t C

(78).

Inclination of two great Circles. Let the equations of these be f A% + Bv + C%=0 (or Ax+By +Cz=0

(79), (80),

A'Z + B'v + C'l;=0 or A'x + B'y + C'z = 0

(81), (82),

and let e, e, have the same values as above, and e', e', corresponding ones. To obtain the inclination of the two circles, we have only, in the formulae given above for the distance of two points, to change p, p', a>, a>', into e, e', e, e'. The distance of a point from a given circle may be found with equal facility; for this is evidently the complement of the distance of the point from the pole of the circle. In like manner we may find the condition that two great circles intersect at right angles, &c. There are evidently a whole class of formulse, not by any means peculiar to the present system of coordinates, such as Ax + By+Cz-s(A'x

+ B'y + C'z)

(83),

for the equation of a great circle subjected to pass through the points of intersection of

Ax + By+ Cz=0, A'x + B'y+G'z = 0. Again,

oo, y, z = 0 a, b, c a', V, c'

(84),

for the equation of the great circle which passes through the points given by the sine systems a : b : c and a' : b' : c', &c, and which are obtained so easily that it is not worth while writing down any more of them. Transformation of Coordinates. Let Xu Ylt Zlt be the new points of reference, and suppose X1 is given by the conjoint systems e = a, b, c. e = a, /3, 7; and similarly Ylt Zx, by the analogous systems e', e'; e", e".

32]

TO THE THEORY OF SPHERICAL COORDINATES.

221

Suppose, as before, P is given by one of the systems a>, p; and let «!, p± be the new systems which determine the position of P with reference to Xl7 Ylt Zx. In the first place, \lt filt vx, are given by the formulae Xl =

7{W(e', e', q) W(e", e", q)} = V{F(e', e', q) F(e", e", q)j _

(85)

'

W(e", e, q) W(e", e, q) = ?", e", q) F (e, e, q)} »/{W(e", e", q) F(e, e, q)} ' F(e, e', q) e, e, q) F ( e ' , e', q)}

F(e, 6^ q) *J{W(e, e, q) F ( e ' , e', q)}'

The system w1 is evidently given immediately by > ^ q)

.

^ ( ^

P>

q)

^(gv.

.

P.

q)

, e, q)} ' J{W(e', e', q)j '
W(e, co, q) . TT(^, «, q) . F ( e " , M, q) e, e, q)} • V I ^ ' , e', q)} • ^{W(e", e", q)}

and from these we may obtain the system px, by means of the formulas *i : 2/i : ^ = 8 ^ + ^

+ 8,^ : k & + W + U i : g^i + f^ + C^

(88).

This requires some further development however. We must in the first place form the system Hi, bu d, fi, gi, Ji: this is done immediately from the formulae of Sect. 2, and we have s'e", e'e", q) kW( e'e", e'e", q) q) W(e", e", q)~ F(e', e, q) F(e", e", q)

bl=

F ( e^e, e^'e, q) kW ( e^e, e^e, q) F(e", e", q)F(e", e, q ) = F (e", e", q) F (e, e, q)'

F(^e',"e?, q)

feF("«',

^?, q)

, e, q) F ( e ' , e', q ) = W(e, e, ^W~(e', e', q)'

7%W, q)

5

e, q)V{F(e', e', q) F(e", e", q)} = F(e, 6, q) V { F ( ^ ' , q)F(e", e", q)}' F ( ee', e'e", q) /;

k W ( ee', e'e", q)

F(e', e', qyVT^"(e Te", q) F(e, e, q)J //

//

& , 6 €,

=

j^\

F(e', e', q) V {W^'Te",
fir)

AJ IT

/

tf

If

\

( e e , e e, q )

F(e", e", q) V{F(e,T q) F(e', e', q)} = F(e", e", q) V {F(e, IT q) F(e', e', q)}"

222

ON SOME ANALYTICAL FORMULA, AND THEIR APPLICATION

*i : S& : *i = V{W(«, ft q)} x

[32 (90),

{Tf (e, p, q) F ( e V [ , eV^, q) + Tf (e', p, q) Tf («V[, ^ e , q) + W(e", p, q) Tf (iV 7 , e?_, q)} :
eV7, ({)+Tf(e') p, q) F ( § [ , ^

q) + TT(e", i>, q) T T ( ^ , <

(})};

these may be reduced to the very simple form , «, q)}

: V{F(e", e", q)} F ( <

«, q).

and in like manner we obtain fix-.yi: *i =
(92),

: V { T f ( 6 ' , 6 ' , q ) } F ( e 5 , p, q), : V{Tf(6", e", q)} F ( P , p, q). It will be as well to indicate the steps of this reduction. Consider the quantity in { } in the first line of the equation which gives the ratios xx : yx : zx; and suppose for a moment e'e" — I, m, n, &c: then, selecting the portion of the expression which is multiplied by a, this is

= al {I K + M + e£) + V (a'? + b'v + c'D + V (o"f + b"v + cTQ], or, since la + l'a' + l"ar = e??,

Ib + I'V + l"b" = 0,

lc + l'c'+ l"c" = 0,

this reduces itself to ee'e". 8^, which is a term of

^ T f ( e V ^ , o, q); and by comparing the remaining terms in the same manner, it would be seen that the whole reduces itself to eeV^TffeV7, «, q); whence the formulae in question. The formulas (86), (87), (91), (92), completely resolve the problem of the transformation of coordinates; they determine respectively px from p or a>, Wj from p or a>. To complete the present part of the subject we may add the following formulae. Suppose

%i : yx : zx = dje1 + alyx + &"z1 : cxx+ c'y^ + c"zlt

(93),

32]

TO THE THEORY OF SPHERICAL COORDINATES.

223

which we see from the preceding formulae is the form of the relation between the systems pt and p. And suppose, as before, X1; /*i, vx are the cosines of the distances of the new points of reference Xl7 Y1( Zx. We can immediately determine the relations that must exist between these coefficients, in order that they may actually denote such a transformation. For this purpose write (94), a , b , c =j a', b ' , c ' = / , a", b", c"=j". Then the distance between the point P and any other point P' is given by the formula cosg_

V {W(p, p, q) W(p', p', q)}

<J{W(Pl,

Pl,

©) W(Pl, Pl', ©)} -

where

e=F(j,j,q),

W(j',f,q),

W(j",j",q), W(j',j",<0. TT(j",j,q), W(j,f, q)...

But we must evidently have COS O =

{pu pi, {W(p,, Ih, qO ^(ft', PL, qi)}

or the quantities © must be proportional to the quantities q, i.e. W(j>j, q) = W(f,f,

q) : W (j", j " , q) : Tf (/, j " , q) : F (/", j , q) : W(j",j, q)

=1 : 1 : 1 : X, : : *,...(98). Ml And in precisely the same manner, if instead of x, y, z, xlt ylt zlt in the above formulae, we had had £, 77, £ : %ly i)lt f1( the result would have been W(j, 3, 4) = ^ (/. / . I) = Tf (/', j " , (J) : F ( / , j " , q) : W(j", j , q) : W (/', j , q) =a : b : C : f : g : f ) ...(99). It is hardly necessary to remark, that throughout the preceding formulae an expression, such as W(p, p', q), is proportional to either of the quantities and may be changed into one of these multiplied by an arbitrary constant, which may be always made to disappear by a corresponding change in another quantity of the same form: thus, for instance, W (p, P', g) x'Z + y'v + *K xt + yv+zS W(p,p,q) but these forms being unsymmetrical, it is better in general not to introduce them. All the preceding expressions simplify exceedingly, reducing themselves in fact to the ordinary formulas for the transformation of rectangular coordinates in Geometry of three dimensions, for the case where the triangle XYZ has its sides and angles right angles. As this presents no difficulty, I shall not enter upon it at present.

224

[33

33. ON THE EEDUCTION OF ~ ,

WHEN U IS A FUNCTION OF

THE FOURTH ORDER. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 70—73.] IT is well known that the transformation of this differential expression into a similar one, in which the function in the denominator contains only even powers of the corresponding variable, is the first step in the process of reducing j-jjj to elliptic integrals. And, accordingly, the different modes of effecting this have been examined, more or less, by most of those who have written on the subject. The simplest supposition, that adopted by Legendre, and likewise discussed in some detail by Gudermann, is that u is a fraction, the numerator and denominator of which are linear functions of the new variable. But the theory of this transformation admits of being developed further than it has yet been done, as regards the equation which determines the modulus of the elliptic function. This may be effected most easily as follows. Suppose U=a

+4
+6cu2

+ 4
i

3

P =aoe + %a?y + %cot?tf + idxy

Also let

+eu\ + ey1.

r = a V4 + 46 V y + 6cVy 2 + 4
be what P becomes after writing x = \x' + fiy' , y = \x' + fiy': and let U' = a' +4&V

+6cV 2

+id'u's

+eV 4

33]

ON THE REDUCTION OF du + JU, &C.

225

Suppose, moreover, k

= \fi/

—Xjfi,

+3c 2 ,

I =ae

-ibd

I'=a'e'

-Wd' + Sc",

J =ace

- ad2 - be* - c3 + 2bcd,

J' = a'c'e' - a'd'» - b'e'* - c'3 + 2b'c'd'; we have evidently xdy — ydx = h (x'dy' — y'dx'), xdy — ydx

or

, x'dy' — y'dx'

Hence writing y uU

~x'

u'-yU

•

~x"

and therefore xdy — ydx

du

x'dy' — y'dx'

du'

we obtain du _ , du' =

the equation between u and u' being

7u u—

Next, to determine the relations between the coefficients of U and ZF. Since P, P' are obtained from each other by linear transformations (Math. Journal, vol. iv. p. 208), [13, p. 94], we have between the coefficients of these functions and of the transforming equations, the relations F = whence also Is' Suppose now U' =-- a' ( H-pu'*)(l+q b' = 0, d' = 0, 6c' = a' (p + g1),

or whence also

/' JI

c.

=

J-a'2 i

a

{f

+ q2

+ 1429gr),

2 's(p + c ?)(34po-»

= a'pq;

226

ON THE REDUCTION OF du + jU,

(p + q) (Mpq - f-?)

WHEN

= 216 ^-3J;

therefore (p+g)2 ( 34 pg - p2 -
27J2 I3 '

which determines the relation between p and q. Also k _ /p2 + g2 + so that _ (f + g3 +

12/ J

~l

If in particular p = — 1, writing also — q for q, du _ /g2 + 14g +1\*

12/ J ~

VU~\ where

1 0 8 g ( 1 - g) 4

=1_

I3

•

Suppose, for shortness^,

l

then

(g2 + 14g+l) 3 -161fg(g-l) 4 = 0,

4 (0-1)*'

Let

g* - g~* =

then

6>3-ilf(6>-l)=O,

which determines 0. And then 7 + 0 + 4(3 + 0)*

[33

33]

U IS A FUNCTION OF THE FOURTH ORDER.

227

Suppose q = a. is one of the values of q; the equation becomes (g2 + 14q + I) 3 _ (a2 + 14a + I) 3 q(q-iy ~ a (a-I)4 ' Now if

_ (/38 + 14/34 + I) 3 /3 4 (/3 4 - I) 4 '

.f 1

_

g = 1-/3V

then which values satisfy the above equation: hence also, identically,

or the values of q take the form

«4 1

^iz^Y

^1±^Y

A-AY

f1 + ^'

(Comp. Abel. (Euv. torn. I. p. 310 [Ed. 2, p. 459].) The equation

&>-M6 + M=0

has its three roots real if 27 — 4<M is negative, and only a single real root if 27 — 41/ is positive. Writing the equation under the form (61 + 3)3 - 9 (0 + S)' + (27 - M) (0 + 3) - (27 - *M) = 0, we see that in the former case 8 has two values greater than — 3, and a single value less than — 3. Writing the equation under the form (6>-l) 3 +3(6»-l) 2 + ( 3 - i f ) ( ( 9 - l ) + l = O, (3 —Mis negative) the positive roots are both greater than 1. Hence, in this case, q has four positive values and two imaginary ones. In the second case 6 has a single real value, which is greater than — 3 and less than 1. Hence q has two negative values and four imaginary ones. In the former case, Is — 27J2 is positive, and the function U has either four imaginary factors or four real ones. In the second case, I3 — 27./2 is negative, or the function U has two real and two imaginary factors.

29—2

228

[34

34. NOTE ON THE MAXIMA AND MINIMA OF FUNCTIONS OF THREE VARIABLES. [From the Cambridge and Dublin Mathematical Journal, vol. i. (1846), pp. 74, 75.]

IF A, B, C, F, G, H, be any real quantities, such that BC + CA + AB - F* - G* - H\ and

(A + B + C) (ABC - AF* - BG2 - CH* + 2FGH)

are positive; the six quantities BC-F\ CA~G\ AB-H\

AK, BK, CK,

(where K = ABC-AFi-BGi-CHS! + 2FGH) are all of them positive. It is unnecessary to point out the connection of this property with the theory of maxima and minima. To demonstrate this, writing as usual BC-F*=A',

GH-AF=F',

CA-G*=B', HF-BG = G', -H* = C, FG and K as above: then if A", B", C", F', G", E", K' be formed from A', B', C, F, 0\ H', as these and K are from A, B, C, F, G, H, we have the well-known formulas A" = KA, F" = B" = KB, 0" = KG, C" =KC, H" = KH,

34]

NOTE ON THE MAXIMA AND MINIMA OP FUNCTIONS OP THEEE VABIABLES. 229

It is required to show that if A' + B' + C and A" + B" + 0" are positive, A', B', C, A", B", G" are so likewise. Consider the cubic equation (A'-k)

(B' - k) (C -k)-

(A' -k)F2-

(F - k) G'* - (C - k) H'2 + 2F'G'H' = 0,

the roots of which are all real. By the formulas just given this may be written k3 - ¥ {A' + B' + C") + k(A" + B" + C")

-K2=0;

and the terms of this equation are alternately positive and negative; i. e. the roots are all positive. Hence the roots of the limiting equation (B'-k)(C'-k)-F'* = 0 are positive, i. e. B' + C and B'C are positive: but from the second condition B', C are of the same sign: consequently they are of the same sign with B' + C, or positive. Also A" = B'G' — F'2 is positive. Similarly, considering the other limiting equations, A', B', C", A", B", C" are all of them positive. In connection with the above I may notice the following theorem. The roots of the equation (A - ka) (B - kb) (G - ok) -(A-

ka) (F - kff -(B-

kb) {G'~ kgf -{G- kc) (H - khf + 2(F- kf) (G - kg) (H - kh) = 0,

are all of them real, if either of the functions Ax" + By2 + Gz2 + 2Fyz + 2Gxz + 2Hxy, ax2 + by2 + cz2 + 2fyz + 2gxz + 2hxy, preserve constantly the same sign. The above form parts of a general system of properties of functions of the second order.

230

[35

35. ON HOMOGENEOUS FUNCTIONS OF THE THIRD ORDER WITH THREE VARIABLES. [From the Cambridge and Dublin Mathematical Journal, vol. i. (1846), pp. 97—104.] THE following problem corresponds to the • geometrical question of determining the polar reciprocal of a plane curve of the third order: the solution of it is also important, with reference to the linear transformations of homogeneous functions of three variables of the third order; reasons for which it has appeared to me worth while to obtain the completely developed result. Let 3U= ax3 + bf + czs + 3iy*z + 3jVte + Skx2y + Sitfz2 +tyxzst?+ Skxxy2 + Uxyz

(1).

It is required to eliminate x, y, z, X from the equations U=0

(2),

¥ + *, = <>,.

(3). w

dy az

)

From the equations (2), (3), we obtain immediately © = %x + 7jy + %z = 0 and thence

(4);

35]

231

ON HOMOGENEOUS FUNCTIONS OF THE THIRD ORDER &C.

so that a single equation more, such as

(6), where <3> is homogeneous and of the second order in x, y, z, would, in conjunction with the equations (3) and (5), enable us to eliminate linearly the seven quantities x2, ?/2, z'2, yz, zx, xy, X. Such an equation may be thus obtained. Let L, M, N, R, 8, T, be the second differential coefficients of U, each of them divided by two. The equations (3) may be written Lx + Ty +8z + X£ = 0,

(7).

And joining to these the equation (4), we have, by the elimination of x, y, z, in so far as they explicitly appear, and X, an equation = 0 of the required form. Hence we may write L,

T,

8,

£

T,

M,

R,

v

8,

R,

JT,

£

(8);

£> v , £ . or substituting for L, M, N, R, S, T, and expanding, (9), * = Ax2 + Bf + Cz2 + 2Fyz + 2Gzx + 2Hxy where the values of A, B, C, F, G, H are (10). [I omit these values (10), and the values in the subsequent equations (13), (14), (20): the values (10) and (13) serve for the calculation of (14), FU, the expression for which with the letters (a, b, c, /, g, h, i, j , k, I) in place of (a, b, c, i, j , k, ^, j l t kx, I) is reproduced in my " Third Memoir on Quantics," Phil. Trans, vol. CXLVI. (1856) pp. 627—647: the values (20) serve for finding that of (21), K(U), but the developed expression of this has not been calculated.] Performing the elimination indicated, the result may be represented by a , h, J > I, fa k, 5 = 0 k , b, ii, i, I, h, V I, ?i > i , c, ii, j ,

2fc

£,

.

2V

.(11).

V

K V

A

B

G F O H

Partially expanding,

(12).

232

[35

ON HOMOGENEOUS FUNCTIONS OF THE THIRD

The values of the coefficients a, b, c, f, g, h may be useful on other occasions: they are as follows Substituting these values the result after all reductions becomes 0 = F(U) =

(14).

It would be desirable, in conjunction with the above, to obtain the equation which results from the elimination of x, y, z from the equations dU „ _„, dx

dU ,=0, dy

^ = 0 dz

.(15),

(i.e. the condition of a curve of the third order having a multiple point), but to effect this would be exceedingly laborious. The following is the process of the elimination as given by Dr Hesse, Crelle, t. XXVIII. (and which applies also to the case of any three equations of the second order). Forming the function VCT, of the third order in x, y, z, by means of the equation ~~ L, T, 8 (16), T, M, 8, (L, M, N, R, S, T, the same as before).

R

R,

N

Then, in consequence of the equations (15), we have not only which is very easily proved to be the case, but also dx

'

dy

.(18),

' dz

as will be shown in a subsequent paper " On Points of Inflection." written.]

[I think never

And from the six equations (15), (18), the six quantities a;2, y2, z\ yz, zx, xy, may be linearly eliminated: we have V U= Aa? + By* + Cz* + 2,1 fz + 3Jz2x + 3Ka?y + Styz2 + SJ^x2 + SK^y* + QAxyz.. .(19), where the values of the coefficients A, B, ... A are (20), and the result of the elimination is K(U)=

a , k,,

j

k ,

o ,

\,

ji ,

i

A

{K(U)

, I,

» c ,

rr

T

XX,

B

Ju

I ,

,

j

k

l t

= 0

(21).

i , I , kx h,

j ,

T

T

2l,

J. y

ij

G,

Ilt

J,

I IT

,

J\. I

L

is consequently, as is well known, a function of the twelfth order in a, b, c,

I, J, K, 1*1, Ji,

Klt

4|.

35]

233

ORDER WITH THREE VARIABLES. The equation

VU=0,

combined with that of the curve, determine, as Dr Hesse has demonstrated in the paper quoted, the points of inflection of the curve. It may be inferred from this, that if U reduce itself to the form U= (aa? + j3y2 + yz* + 2iyz + 2/cxz + 2Xxy) P = VP.

.(22),

P a linear function of x, y, z: then VU takes the form .(23),

V U = P (pV +
where p is of the second order in the coefficients of P, and also in the coefficients a, /3, 7, t, K, X: and u is equal to the determinant

multiplied by a numerical factor.

a,

X, K

\

A

i

K,

i ,

7

(24),

If U is of the form U = PQR

then

(25),

VU=pPQB = PU

(26),

and this equation is consequently the condition of the function IT being resolvable into linear factors. The equation in question resolves itself into A B _G a "b~c"

A i

j

k

ix

h

a system which must contain three independent equations only. to verify this & posteriori.

C.

(27); It would be interesting

30

234

[36

36. ON THE GEOMETEICAL EEPEESENTATION OF THE MOTION OF A SOLID BODY. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 164—167.] LET P, Q, R, be consecutive generating lines of a skew surface, and on these such that pq1, qr' are the shortest distances take points p', p; q1, q; r', r between P and Q, Q and R, &c. Then for the generating line P, the ratio of the inclination of the lines P, Q to the distance pq' is said to be "the torsion," the angle q'pq is said to be the deviation, and the ratio of the inclination of the planes Qpq' and Qqr' to the inclination of P and Q is said to be the "skew curvature." And similarly for any other generating line; so that the torsion and deviation depend on the position of the consecutive line, and the skew curvature on the position of the two consecutive lines. The curve pqr is said to be the minimum distance curve [or curve of strictionj. {When the skew surface degenerates into a developable surface, the torsion is infinite, the deviation a right angle, the skew curvature proportional to the curvature of the principal section, i.e. it is the distance of a point from the edge of regression, multiplied into the reciprocal of the radius of curvature, a product which is evidently constant along a generating line. Also the curve of minimum distance becomes the edge of regression.} A skew surface, considered independently of its position in space, is determined when for each generating line we know the torsion, deviation, and skew curvature. For, assuming arbitrarily the line P and the point p, also the plane in which pq' lies, the position of Q is completely determined from the given torsion and deviation; and then Q being known, the position of R is completely determined from the skew curvature for P, and the torsion and deviation for Q; and similarly the consecutive generating lines are to be determined. Two skew surfaces are said to be "deformations" of each other, when for corresponding generating lines the torsion is always the same. Thus a surface will be deformed if considering the elements between the successive generating lines P, Q as rigid, these

36]

ON THE GEOMETRICAL REPRESENTATION OF THE MOTION OP A SOLID BODY.

235

elements be made to revolve round the successive generating lines P, Q and to slide along them. {They are " transformations", when not only the torsions but also the deviations are equal at corresponding generating lines: thus, if the sliding of the elements along P, Q be omitted, the new surface will be, not a deformation, but a transformation of the other.} No two skew surfaces can be made to roll and slide one upon the other, so that their successive generating lines coincide, unless one of them is a deformation of the other: and when this is the case, the rolling and sliding motions are completely determined. In fact the angular velocity of the generating line is the angular velocity round this line, into the difference of the skew curvatures of the two surfaces; the velocity of translation of the generating line in its own direction is to the angular velocity of the generating line, as the difference of the deviations is to the torsion. {This includes also the case in which one surface is a transformation of the other, where the motion is evidently a rolling one.} A skew surface moving in this manner upon another of which it is the deformation, may be said to " glide" upon it. We may now state the kinematical theorem: " Any motion whatever of a solid body in space may be represented as the ' gliding' motion of one skew surface upon another fixed in space, and of which it is the deformation." a theorem which is to be considered as the generalization of the well-known one— "Any motion of a solid body round a fixed point may be represented as the rolling motion of a conical surface upon a second conical surface fixed in space." and of the supplementary theorem— " The angular velocity round the line of contact (the instantaneous axis) is to the angular velocity of this line as the difference of curvatures of the two cones at any point in the same line, to the reciprocal of the distance of the point from the vertex." The analytical demonstration of this last theorem is rather interesting: it depends on the following formulae. Forming two determinants, the first with the angular velocities round three axes fixed in space, and the first and second derived coefficients of these velocities with respect to the time; the other in the same way with the angular velocities round axes fixed in the body; the difference of these determinants is equal to the fourth power of the angular velocity into the square of the angular velocity of the instantaneous axis. To show this, let p, q, r be the angular velocities round the axes fixed in the body; u, v, iv those round axes fixed in space; to the angular velocity round the instantaneous axis; V, 12 the two determinants: the theorem comes to

v - n = M, where

2

s

2

2

M = OJ (p' + g' + r' - a>'2), or w2 (it'2 + v1* + w'2 - «' 2 ).

Here

u = ap + ft q +yr, v ^rfp+ffq+y'r, w = al'p + ff'q +
236

ON THE GEOMETRICAL REPRESENTATION OF THE MOTION OF A SOLID BODY. [ 3 6

whence

u' = a p' + /3 q1 + y r',

w' = al'p' + /3'V + y"r', (the remaining terms vanishing as is well known); and therefore vw' — v'w = a (qr' — q'r) + /3 (rp' — r'p) + y (pq1 — p'q), wv! — w'u = a' (qr' — q'r) + /3' (rp' — r'p) + y' (pq' — p'q), uv' - u'v = a" (qr' - q'r) + /3" (rp' - r'p) + 7" (pq' — p'q). Hence vw" - v"w = a (qr" - q"r) + /3 (rp" - r"p) + y (pq" -p"q) + u'co2 - uwco', wu" - w"u = d (qr" - q"r) + /3' (rp" - r"p) + y' (pq" - p"q) + v'a2 - veoco', uv" - u"v = a" (qr" - q"r) + j3" (rp" - r"p) + y" (pq" -p"q) + w'eo* - wcoa>', and multiplying these by u', v1, w', and adding, the required equation is immediately obtained. In fact, if r be the distance of a point in the instantaneous axis from the vertex, and p, o- the radii of curvature of the two cones at that point, then r

to3 ,~

r

a) _

— = —-. V

- = — j 12>,

as may be shown without difficulty: and the angular velocity of the instantaneous axis if* is given by the equation vs = —-; hence the relation between the two angular velocities is to : w =

1 p

1 a

1 :—. r

37]

237

37. ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [From the Cambridge and Dublin Mathematical Journal, vol. i. (1846), pp. 167—173 and 264—274.] THE difficulty of completing elegantly the solution of this problem, in the case where no forces act upon the body, arises from the complexity and want of symmetry of the ordinary formulas for determining the position of one set of rectangular axes with respect to another set; in consequence of which it has hitherto been considered necessary to make a particular supposition relative to the position of the fixed axes in space, viz. that one of them shall be perpendicular to the "invariable plane" of the rotating body. But some formulae for the above purpose, given also by Euler, are entirely free from these objections. Imagine two sets of axes Ax, Ay, Az, Ax/t Ayp Azr The former set can be made to coincide with the second set, by a rotation 6 round a certain axis AR, inclined to Ax, Ay, Az at angles f, g, h. (As usual f, g, h are the angles RAx, RAy, RAz considered as positive, and the rotation is in the same direction as a rotation round Az from x towards y.) This axis may be termed the resultant axis, and the angle 6 the resultant rotation. The formulae of Euler express the coefficients of the transformation in terms of the resultant rotation and of the position of the resultant axis, i.e. in terms of 6 and of the angles /, g, h, whose cosines are connected by the equation cos 2 /+ cos2 g + cos2 h = 1. This idea was improved upon by M. Rodrigues (Liouv. torn. v. p. 404), who introduced the quantities tan \ 8 cos/, tan \ Q cos g, tan \ 6 cos h, (quantities which will be represented by X, /x, v) by means of which he expressed the

238

ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT.

[37

coefficients as fractions, the numerators of which are very simple rational functions of the second order of X, fi, v, and which have the common denominator (1 + X2 + fjp + v2). These quantities may conveniently be termed the "coordinates of the resultant rotation," and the denominator or the square of the secant of the semi-angle of resultant rotation will be the "modulus" of the rotation. The elegance of these results led me to apply them to the mechanical question, and I gave in the Journal (vol. in. p. 224), [6], the differential equations of motion obtained in terms of X, /*, v: which I integrated as in the common theory, by supposing one of the fixed axes to be perpendicular to the invariable plane. Though my attention was again called to the subject, by the connexion of some of these formulae with Sir William Hamilton's theory of quaternions, no other way of performing the integration occurred to me. The grand discovery however of Jacobi, of the possibility of reducing to quadratures the two final differential equations of any mechanical problem, when the remaining integrals are known, induced me to resume the problem, and at least attempt to bring it so far as to obtain a differential equation of the first order between two variables only, the multiplier of which could be obtained theoretically by Jacobi's discovery. The choice of two new variables to which the equations of the problem led me, enabled me to effect this with the greatest simplicity; and the differential equation which I finally obtained, turned out to be integrable per se, so that the laborious process of finding the multiplier became unnecessary. The new variables il, v have the following geometrical interpretations, 12 = k tan \ 6 cos I, where h is the principal moment, 6 as before the angle of resultant rotation, and / is the inclination of the resultant axis to the perpendicular upon the invariable plane, and v = k2cos2^J; where, if we imagine a line AQ having the same position relatively to the axes in fixed space that the perpendicular upon the invariable plane has to the principal axes of the rotating body, then J is the inclination of this line to the above perpendicular. To the choice of these variables I was led by the analysis only. It will be seen that p, q, r are functions of v only, while X, /t, v contain besides the variable XI. In obtaining these relations a singular equation O2 = KV — k1 occurs (equation 13), which may also be written 1 + tan2 \ 6 cos2 / = sec2 £ 0 cos2 £ J, in which form the interpretation of the quantities / , J has just been given. The equation (17), it may be remarked, is self-evident: it expresses that the inclination of the resultant axis to the normal of the invariable plane, is equal to the inclination of the same axis to the line A Q. Now the resultant axis having the same inclination to the axes fixed in space as it has to the principal axes, and the line AQ the same inclinations to these fixed axes that the normal to the invariable plane has to the principal axes, the truth of the proposition becomes manifest. The correspondence in form between the systems (10) and (14) is also worth remarking. The final results at which I arrive are, that the time and the arc whose tangent is fl -r- k, are each of them expressible as the integrals of certain algebraical functions of v. The notation throughout is the same as that made use of in the paper already quoted. The equations of rotatory motion are A*

— dp _ dq _ dr _d\ _d/j, _ dv

.

37]

ON THE ROTATION OP A SOLTD BODY ROUND A FIXED POINT.

239

where

(2).

A = 1 { ( 1 +X>)p + (xh-v M = -J-

{(/JX

)q + (Xv + fi)r},)\ p + ( 1 + fj?) q + {jw — X) r}! y

+ P)

(3).

2

N = -J- {(i/X - fx) p + (fiv + X) q + ( 1 + v ) r], / [whence also

XA + pM + vN = \ K (Xp + pq + vr)

(3 bis)].

In the case where the forces vanish, the first three equations become simply

Q

1 =B

R =

(4).

C

and here the usual four integrals of the system are Ap2 + Bq2 +Cr* = h 2

2

(5),

2

Ap (1 + X - p - v ) + 2Bq (X/A (6), 2

2Ap (v\ -fi) + 2Bq (jiv + X) + Cr (1 + v* - \ - ^

2

2

= c (1 + A. + (J? + ^ ), J

or as they may also be written, a (1 + X2 - yw,2 - i/2) + 2 b ( V + v) + 2c (VX - p) = Ap (1 + X2 + yx2 + v2), ^ 2a(X/JL-v)+h(l

+ /^-vs-X2)

+ 2c(^ 2

+ X) = Bq(l+X? + fJJi + v% [ 2

(6 bis);

2

2a (vX + [*) + 2b (JJ,V - X) + c (1 + v - X - /u?) = Cr (I + X + ^ + v*), to which we may add, •(7); where

2

2

2

3

k =a +b +c

..(8).

Introducing the quantities K, 12, (the former of which has been already made use of) given by the equations 2 K = 1 + X + fj + v\ ) .(9). O = \Ap + fiBq + vCr, j

240

ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT.

[37

The equations (6) may be written under the form 2\ft + 2/j.Cr - 2vBq = K (Ap + a) - 2Ap," + 2vAp = K (Bq+h) - 2Bq, >

-2\Cr+2/j,n

(10),

2\Bq - 2fiAp + 2vD, =K(Cr + c) - 2Cr ,) whence also, multiplying by Ap, Bq, Cr, and adding, 2ft2 = K {t? + (Apa, + Bqb + Ore)} - 21?

(11),

or writing 2v

(12),

this becomes D? = Kv-k*

(13);

an equation, the geometrical interpretation of which has already been given. From the equations (10) we deduce the inverse system aft - bCV + cBq = 2\v - ClAp," &Cr +bn - cAp = 2fiv - tlBq, i-

(14),

- &Bq + bAp + eft = 2vv - ftCr , J which are easily verified by multiplying by ft, Cr, — Bq; or by — Cr, ft, Ap; or Bq, — Ap, ft: adding and reducing, by which means the equations (10) are re-obtained. Hence also if for shortness V =&qr(B-C) + brp(C-A)+cpq(A-B),j we have, multiplying by p, q, r, and adding, ft3> -V = 2v(\p + jiq + vr) - ilh

(16).

To these may be added the equation ft = a\ + b/i + cv

(17),

which follows immediately from either of the systems (10) or (14). We may also put the equations (10) under this other form, 2Xft - 2/iC + 2i>b = K (Ap + a) - 2a,' 2Xc + 2/xft - 2i>a = « (.% + b) - 2b, >•

{10 bis).

- 2Xb + 2fia, + 2z>ft = K(CV + c) - 2c, J It may be remarked now, that p, q, r are functions of v; since we have to determine these quantities, the three equations Ap2 + Bq2 + Cr2 = h, a

2

"j

1

J p + .BV + CV = # ,

>• 2

Apa, + Bqb + Crc =2v- k , j

(18).

37]

ON THE ROTATION OF A SOLID BODY BOUND A FIXED POINT.

241

Also \, fi, v are given by the equations (14) as functions of p, q, r, fl, i.e. of v, O,; so that every thing is prepared for the investigation of the differential equation between v, 12. To find this we have immediately dv = §(AsLdp + Bbdq + Ccdr) = iVdt

(19),

from the equations (4) and (15). V is of course to be considered as a given function of v. Again, (20), where

die = 2 (\dX + fidfi + vdv)

(21);

or from the equations (1), (3), [and (3 bis)], d/c = /c (Xp + /jq + vr)dt

(22).

Hence, from (16), 2vdK = ie{Q,(h + ®)-V}dt or

2 (vd/c + icdv) = icQ, (h + <&) dt

whence

(23); (24),

dQ, = \ K (h + <E>) dt,

= i=-±^(A + *)(ft

(25),

and therefore, from (19), 2dn

h + <&, =-*-*»

. . '

(26)

functions of p, q, r by the equations (15), and these quantities are functions of v by (18). The variables in (26) are therefore separated, and we have the integral equation

where 8 is the constant of integration.

The equation (19) gives also (28);

and thus the solution of the problem is completely effected. The integrals may be taken from any particular value v0 of v. The variable Q may be exhibited as the integral of an explicit algebraical function, by recurring to the variable 0 of the paper quoted. Thus if Ap? + Bqf + CV02 = h, AW + &qf + Or? = k\ Ap
31

242

ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT.

[37

then the values of p, q, r are respectively

where pqr

2dv V

dv or -= = V V pqr'

and then

in which form it is exactly analogous to the equation there obtained, p. 230, [6, p. 34] ,, , Uh + kr)dj> 4 tan-1 v0 = f. n { T . J(k + Cr)pqr

On the Variation of the Constants, when the body is acted upon by Forces. The dynamical equations of a problem being expressed in the form dLdT_dT=clV dt d\' dX~ d\' dt d/i'

d/M d/i '

ddT_dT_dV dt dv' dv dv suppose the equations obtained from these by neglecting the function V, are integrated; each of the six integrals may be expressed in the form a = / ( \ , fi, v, X', p, v, t), where a denotes any one of the arbitrary constants. Assume dT dX

dT dfi

dT dv

then X', fi, v may be expressed in terms of \, /*, v, u, v, w, and the integrals may be reduced to the form a — F(\, fi, v, u, v, w, t). These equations may be considered as the integrals of the proposed system, taking into account the terms involving V, provided the constants [say a, b, c, d, e, / ] be sup-

37]

ON THE EOTATION OF A SOLID BODY BOUND A FIXED POINT.

243

posed to become variable. We have, in this case, by Lagrange's theory of the variation of the arbitrary constants, the formulae da

,

j,dV

,

sdV

.

.

,,dV

, dV

,

,.dV

where t

XN

'

(da db \du dX

da db\ dX du]

Ida db da db\ \dv d/n dfj, dvj

Ida db da db\ \dw dv dv dw) '

and in which V is supposed to be expressed as a function of a, b, c, d, e, f, t. Thus the solution of the problem requires the calculation of thirty coefficients (a, b), or rather of fifteen only, since evidently (a, b) = — (b, a). It is known that these coefficients are functions of a, b, c, d, e, f, without t; so that, in calculating them, any assumed arbitrary value, e. g. t = 0, may be given to the time. In practice, it often happens that one of the arbitrary constants, e.g. a, may be expressed in the form a = F(X, (i, v, u, v, w, t, b, c, d, e, f), where b, c, d, e, f are given functions of X, p, v, u, v, w, t. seen that we may write

In this case, it is easily

(a, &)-{(«, 6)} + (o,6)g + (ct6)g + ( ^ ) g + (/,6)|, •vhere, in the calculation of {(a, b)}, the differentiations upon a are performed without
2 K

Ap-

vBq + fiCr),

(29),

vAp + Bq- XCr),

2

w = - (— iiAp + \Bq + Gr) ; equations which may also be expressed in the form 2Ap = ( 1 + \ 2 ) u + (X/t + v) v + (vX - fi) w

(30),

2Bq = (\p - v) u + ( 1 + fi2) v + (jw + X) w, 2Cr = (vX + fi) u + (jiv - X) v + ( 1 + i>2) w, or putting for shortness Xu + /J,V + vw = in

(31), 31—2

244

ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT.

these become

2Ap = Xm + u+vv — pw,

[37 (32).

2Bq = fitn — vu+ v + Xw, 2Cr = vor + fill — Xv + w, whence also 2O=«OT

(33).

Substituting the values of Ap, Bq, Cr, given by (30) in the equations (6), we deduce 2a = XOT + u — vv + /xw

(34),

2b = /Ltnr + vu+ v — Xw, 2c = vis — fiu + Xv + w, whence also 2 (aX + b/i + cv) = Km

(35),

which in fact follows from (33) and (17). And likewise the inverse system, 2 fC

a + vh—fic)

(36).

2 v = - (— z/a + b + Xc), K

2 w = - ( /u.& — Xb + c). fC

It is easy to deduce (37), *]

(38).

Again, from the equations (10 bis), K.

(hCr - cBq) = - 2X (a2 + b2 + c2) + 2a (Xa + fib + vc) + 2 (bv - Cfi) O = - 2XP+ 2 (a + bv-c/j) X2 = - 2Xt +

and, forming also the similar expressions for K (cAp — aCV), and K (stBq — bAp), we thus obtain fC

]?X

b C B

(39),

2 D,v — k2fi — cAp — bCr, fC

2 — k^v = &Bq — cAp; fC

to which many others might probably be joined. The constants of the problem are a, b, c, h, e, B. Of these a, b, c are given as functions of X, fi, v, u, v, w, by the equations (34); in which OT is to be considered as

37]

ON THE ROTATION OF A SOLID BODY BOUND A FIXED POINT.

245

{These determine k2, which is however given immediately by

standing for Xu + fiv + vw. (37).} As for h, we have

where Ap, Bq, Gr are given as functions of X, ft,, v, u, v, w by (32), in which also stands for Xu + /xv + vw. Again,

2k ~

3

in each of which V, <E> are functions of v, and of a, b, c, h, partly as entering explicitly into these functions, partly as contained implicitly in p, q, r, which enter into V, <3>, and are functions of v, h, k given by (18). After the integration v is to be considered a function of X, //,, V, U, V, W given by (38). Both of the integrals may be supposed taken from a certain value v0 of v, which may be considered as an absolutely invariable arbitrary constant, since without it we have the right number, six, of arbitrary constants. First to find (a, b), (b, c), and (c, a). From (34) we have

+ (Xfi-v)

(fiv + VT) - (Xv + w) ( 1 + /t2)

+ (vX + /"•)( u + fiw) — (Xw — v) (fiv — X)} = •£ {fiu — Xv — w — vvi) = — \ 2c = — c; whence the system (b, c) = - a , Also we may add

(c, a) = - b ,

(a, b) = - c

(40).

(k, a) = r (a, a) + -= (b, a) + r (c, a) = 0,

or (k, a) = 0,

(k, b) = 0,

(k, c) = 0

(41),

which will be useful in calculating some of the following coefficients. Proceeding to calculate (a, h), (b, h), (c, h). It is seen immediately that (a, h) = 2 {p (a, Ap) + q (a, Bq) + r (a, Gr)}, where Ap, Bq, Gr, are given by the equations (32), so that (a,Ap) = i{

(l+X2)(Xu+

w ) - ( l +X2)(Xu+

w)

i — v ) (Xv — w)— (Xfi + v) (Xv + w) v + Xw) — (vX — ju.) (— v + Xw)} (42).

246

ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. Similarly

[37

( 1 + X2) (/M + w) - ( Xu +CT)(Xfj, - v)

(a, Bq) = \{

+ (Xfi - v ) (fiv + vr) - ( Xv + w) ( 1 +/i 2 ) + (vX + fi ) (fiw — u) — (—v + Xw) (fiv + X )} i.e.

(a, Bq) = O,

(43),

and similarly

(a, Gr) = 0 ;

whence

(a, h ) = 0, and therefore (b, h) — 0, (c, h) = 0

also

(Jc, h)=0,

(44); (45).

Next we have to determine (a, e), (b, e), (c, e). Here e being a function of u, v, w, X, fi, v, a, b, c, h, we must write (a, e) = {(a, e)} + (a, b) -=r + (a, e) -3- + (a, h) 37 , i.e.

(a)e)={(a,e)} + b But

de J c

e = t — 2 | = ; and thence {(a, e)} = — =• (a, v),

and v is given immediately as a function of X, fi, v, u, v, w, by the equation (38). Hence (a, v) = \[ ( 1 + X2) {(1 + K) um + Xt*2} - ( XM + is) {u + X (1 + K) OT} + (Xfj, - v ) {(1 +K)VW

+ fim2} -(Xv+

w) {v + fi (1 + K) iff}

2

+ (vX + fi) {(1 + K) Wm + via )} — (—v+ Xw) {w + v (1 + K) T&}] = I {(1 + K) T3U - X (1 + K) ZJ2 + XK - X (W2 + V2 + W2) - IMS] = \ {KWff - XCT2 - X (u2 + v*+ w2)}

j

{by (37) and (33)}, (46);

whence

{(a, e)} = — ^ (bGr — cBq), (a, e )

=

--(b0r-c5g)+b^-c^.

The terms b -=— c-=• are evidently of the form F(v)—F(v0). If therefore we suppose v = v0, we have (a, e) = - l ( b C V 0 - c % ) V1 0

(47),

37]

ON THE BOTATION OF A SOLID BODY BOUND A FIXED POINT.

247

if p0, q0, r0, Vo refer to the value v0 of v, i.e. if

+ Cr^h p

q

(48),

= 2v0 - h\

{This implies evidently

an equation which it is interesting to verify. ,de ac

de ab

o

f, f, d J \ ac

In fact, from the value of e

d\ 1 ab/V

a

[, 1 A dV dV\ J V2V ac ab/

or we have to show that d 1 ., n dv V

j, , 2/

2 /, dV v 2 V etc

dV\ 2 s a b / V2

if for shortness o = b - r —c-jr •

ac

ab

Now V containing a, b, c explicitly, and also as involved in p, q, r, we have

suppose. The equation to be verified becomes

Now, observing that Bk = 0, we have A'pSp + B2qSq + C*rSr = 0, A aZp + BbSq + GcSr =

-(bCr-cBq).

Also,

dp av

„ , da „ dr . an dv

248

ON THE BOTATION OF A SOLID BODY BOUND A FIXED POINT.

[37

whence evidently dp_ - 2 dv ~ bCr - cBq P'

dq _ - 2 dv ~ bCr - cBq q' dV dv

dr _ -2 dv ~ bCr - cBq

'

-2 .,_ hCr — cBq '

or the equation to be verified is simply

which follows immediately from the three equations just given for the determination of dp dq dr , dv' dv' dv '• From these values also (k, e ) = 0

(49).

Next, to calculate (h, e),

(h, e)={(h, e)} + (h, a)g+(A, b)~+(A, c) | ; but the three last terms being evidently such as to vanish for v = v0, we may neglect them, and consider (h, e) as the value which {(h, e)} assumes for this value of v. Now

{(A, e)} = 2p {(Ap, e)} + 2q {(Bq, e)} + 2r {(Or, e)}, 2 {(Ap, e)} = -^(Ap,

where

v),

and (Ap,

v)=l[

( 1 + X2) {(1 + K) UTS + XOT2} + (\/Jb +

V) {(1 +

IC) VST + flVS2} -

(\u

+

(\V -

«•) {u + \

(1 + K) •&}

W) {V + fl ( 1 + K) XS)

2

+ (v\ - fi) }(1 + K) wzy + i/or } - ( V + \w) {w + v (1 + K) VT}] = I {(1 + K) nu + XKOT2 -X

(1 + K)OT2- X (M2 + v2 + w2) - ™u\ = (a,

v)

(50), whence

{(Ap, e)} = - ^ (hCr - cBq)

and therefore

{(Bq, e)} = - ^ (cAp - &Cr), {(Cr,

e)}=-±

(51),

37]

ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT.

whence

249

{{h, e)j = — 2,

and therefore

(52).

(h, e ) = - 2 Next, to find (a, 8), (b, 8), (c, 8), (h, 8), 8= 2tan

2T-

= 8' + 8" suppose, (a, S) = (a, 8') +(a, 8"), 7

{observing k where (a,

+

X2) (/cv

+

2XOT) -

( \ u + OT) KX

— v) (KV + 2/ATO-) — ( Xtt + W) Kfl

+ (v\ + fi) =|

+ 2v&) — (—v+ Xw) KV }

(KW

XO = -|(a + Ap) K

K(U

(53),

by equations (29), (33), and (10) ; that is

(a, B') = ^-(& + Ap). Zv

Also

(a, S") = -

(a, v) + (a, b) ~

+ &c.

whence (a, 8) = ^

(bCr - cBq)} +Fv-

Fv0,

or putting v = v0, (a, 8) =

(54),

'2v0'

and therefore also

k

(b, 8) = ^ { b + % (c> 8) =

C.

H-

(c +

^

(cAp0 - aOr0)}, - bAp0)}. 32

250

ON THE ROTATION OP A SOLID BODY ROUND A FIXED POINT.

Again,

[37

( h, 8) = 2p(Ap, B) + 2q(Bq, 8) + 2r(Cr, 8), (Ap, S) = (Ap, V) + (Ap, 8"),

(Ap, K-B) = § { ( 1 + X2)

(KU

+ 2\OT) - (\u + w)

K\

+ (Kfi + v) (KV + 2/iTO-) — (\v — w) K/J, + (v\

+ fi) (KW + 2VUT) —(V =%

therefore

+ Xw) KV }

« ( a + Ap)

(Ap, V) =•£-(*+ Ap)

(55);

(56),

Jll

- cBq) + Fv- Fv0,

8)=±{a,-Ap-h^

(Ap,

and similarly for (Bq, S), (Cr, S). Substituting, and neglecting the terms which vanish for v = v0,

(h, 8) = 0

(57).

Lastly, to find (e, 8),

where, in {(e, 8)}, the differentiations upon e are supposed not to affect the constants a, b, c. Neglecting the terms which vanish for v = v0, (e,8)

={(e,8)},

where, in [{(e, B')}], the differentiations upon e and 8 do not affect the constants. da,

37]

251

ON THE ROTATION OP A SOLID BODY ROUND A FIXED POINT.

neglecting the terms which vanish for v = v0, therefore

(e, 8) = [{(e, &')}] + [{(e, 8")}] =[{(«, SO}];

since (58),

Hence + ( ! + «) Xtsr} (2\CT + KM) -

(u,

(1 + K) TXU) K)

+ [W + (1 + K)

K\

1ZV} Kfl

+ ( 1 + K)OTWJKV

(59), therefore (e, 6) = - = -

.(60).

Hence, recapitulating, (b, c ) = - a ,

(c, a ) = - b ,

(a, b) = - c ,

(a,A) = O,

(b,A) = O,

(c,h) = O,

(a,

e)=-~(bCro-cBqt>),

(b, e) = - =• ( c ^ 0 - &Cr0), (c, e) = - =- (a,Bq0 - \>Ap<,), v

o

(h, e) = - 2,

(61),

(cAp» - aOr.)},

(b, 6) = jj—

(c, S) = A ( c + O r 0 - ' (A, S) = 0, (e, « ) =

*

32—2

252

ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT.

[37

and therefore

da db

dl

=

do

1 „ „

dV

1 , .

-^o

+ c

,dV

=

dt

dV . , dV dV

d a - ^

„ , dV . k , .

A

C

n

r

,dV

^

+

k ,,

„

J

h + ®an „

A

h

+

A + *o. . B

{

A

„ „ dV

G

„ ., dV

)

]

dV

_2 de'

{b + Bq0

^—° (cAp0 -aCr 0 )}

to which we may join

dk_dV

*-ds We have thus the complete system of formulae.

(63)

-

253

38]

38. NOTE ON A GEOMETRICAL THEOEEM CONTAINED IN A PAPER BY SIR W. THOMSON. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 207, 208.] IT is easily shown that if three confocal surfaces of the second order pass through a point P , then the square of the distance of this point from the origin is equal to the sum of the squares of three of the axes, no two of which are parallel or belong to the same surface (the squares of one or two of the axes of the hyperboloids being considered negative); i. e. if x2 y2 z2 _

aT+h + ¥Th so2

y2

+

cTVh~ ' z* _

+

aF+k W+lc+ c*+l~ ' a2 + I

then

b2 + I

c2 + I

'

a? + y2 + z2 = a2 + b2 + c2 + h + k +1. In fact these equations give (a 2 -6 2 )(a 2 -c 2 )

2_

and adding these and reducing, we have the relation in question; which is also immediately obtained by forming the cubic whose roots are h, k, I.

254

NOTE ON A GEOMETRICAL THEOREM &C.

[38

From this property may be deduced the theorem given by Mr Thomson in the preceding memoir ["On the Principal Axes of a Solid Body," pp. 127—133 and 195—206, see p. 205]. In fact, writing r 2 = a? + y2 + z2, k - - a? - b2 - c2 + h,

and

we see that in consequence of these relations the equations W /,2

*

<*

r2 + a2-h a?

11 Tk

_

Z ^2

~"~

f

'

*

=1

r> + b2-h r2 + c2-h 2 y z2 + ~— + = 1,

are equivalent to two independent equations, i.e. the third can be deduced from the two first. Now the first equation is that of an ellipsoid (or generally a surface of the second order, since a, b, c are not necessarily real). The second is that of what may be called a conj ugate equimomental surface, defining the term as follows: " The conjugate equimomental surfaces of an ellipsoid (or surface of the second order) -j + fs + - 5 = 1, are the equimomental surfaces derived in the usual manner from any surface of the second order y

2

+

J~-J-2

+T

3

= 1, which is confocal with the conjugate surface ~i + j-2+-2 = — 1

of the given ellipsoid," viz. by measuring along any line through the centre distances equal to the axes of the section by a plane through the centre perpendicular to this line, and taking the locus of the points so determined for the equimomental surface. The third equation is that of a surface confocal with the given ellipsoid; hence the theorem, "The curves of curvature of a given ellipsoid lie upon a system of conjugate equimomental surfaces." But since the first and second equations are evidently satisfied by the combination of the first equation with the relation r2 = h, which is that of a sphere, we have also, " The curve of intersection of the ellipsoid with any one of the conjugate equimomental surfaces, is composed of the line of curvature and a spherical conic." And these two curves being each of them of the fourth order make up the complete curve of intersection, which should obviously be of the eighth order. It would be an interesting question to determine the relations existing between the curve of curvature and the spherical conic, which have been thus brought into connection by means of the conjugate equimomental surfaces; i.e. between the two curves obtained by combining the equation of the ellipsoid with rr&

ftp-

2^

a2 + k + b 2 + k + c 2 + k =

lj

r2 = a2 + 62 + c2 + k, respectively: but it will be sufficient at present to have suggested the problem.

39]

255

39. ON THE DIAMETRAL PLANES OF A SURFACE OF THE SECOND ORDER. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 274—278.] LET U = Aa?+By2 + Gz2+2Fyz + 2Gxz+2Hxy = 0, be the equation of a surface of the second order referred to its centre, and let ax + a'y + a"z = 0 be the equation of one of its diametral planes; then, as usual (A-u)a + Ha'+ Ga" = 0, Ha+(B-u)a' + Fa" = 0, Ga+ Fa'+(C-u)a" = 0, which are equivalent to two independent equations, and consequently capable of determining the ratios a : a' : a", provided that u satisfy the cubic equation that is obtained by eliminating a, a', a" from the three equations. We have from the second and third, from the third and first, and from the first and second equations respectively,

a:a':

«" = » , : ^

:ffi/= ^

: 23, : £, = &, : $, : ffi,;

where, if

m=BC -F* ,

,, 23,, ®,J JP,, ffi,, i^, are what these become when A, B, C are changed into

256

ON THE DIAMETRAL PLANES OF A SURFACE OF THE SECOND ORDER.

[39

A —u, B—u, C — u, so that

<&,=<&-(A+B)u + u\

Hence the equation ax + a!y + OL'Z = 0 may be written in the three forms

or, what comes to the same thing, as follows, u {Ax + Hy+ Oz) + vx = 0, x + By + Fz) + vy = O, + <&z + u {Ox +Fy+ Gz) + vz - 0, in which for shortness v has been written instead of u?-{A+B + C)u. The elimination of u, v from these equations gives a result © = 0, where 0 is a homogeneous function of the third order in x, y, s; and this equation, it is evident, must belong to the three diametral planes jointly, i.e. © must be the product of three linear factors, each of which equated to zero would correspond to a diametral plane. Thus the system of diametral planes is given by

z, Ax + Hy + Oz, x = 0, z, Hx + By+Fz, y z, Gx+Fy+Cz, z or developing the determinant, as follows,

+ { G (CD - 23) - (& {G - B) - (H§ + { H031 - ®) - ft(A- C, - (F® + { F (33 - a ) - JF (B - A) - {Gffi - H<&)} xf + {- H(€ - 23) + n (° ~ Z) + {-F (&-<&) + jF (A-C) + {- 0 (23 - ») + <E {B-A) + (023 - BM + a ® - C^ + s a - ^23) xyz;

39]

ON THE DIAMETEAL PLANES OP A SURFACE OF THE SECOND ORDER.

or reducing ©=

257

{F (G* - J5P) - GH (C - B)} a? + {G (H* - F") -HF{A-G)} ys + {H (F* - G2) -FG(B-A)} zs + {G(A-B)(B-(J) + FH(A+B-2C) + G(F'+G°- 2IP)} yz> + {H(B-G) (C-A) + GF(B+C-2A)+H{G*+H*-2F>)}za? + {F(G-A) (A-B) + GH{G+A - 2B) +F(H* + F*- 2G*)} xf

-{(A-B) {B-C) (CIn the case of curves of the second order, the result is much more simple; we have © = Ax + Hy, Hx+By,

x =0 y

for the equation of the two diameters. The above formulae may be applied to the question of finding the diametral planes of the cone circumscribed about a given surface of the second order, (or of the lines bisecting the angles made by two tangents of a curve of the second order). Considering the latter question first: if

be the equation of the curve, and a, y3 the coordinates of the point of intersection of the two tangents, the equation of the pair of tangents is Va2

o"

I \ai o*

or making the point of intersection the origin,

i. e.

(/3* - ay)2 -(b2a? + ay) = 0;

whence A =fi*— 62, B = a2 — a2, H= — a/3, and the equation to the lines bisecting the angles formed by the tangents is aj3 (of - f) - {a2 - /32 - (a2 - 62)} xy = 0, which is the same for all confocal ellipses; whence the known theorem, "If there be two confocal ellipses, and tangents be drawn to the second from any point P of the first, the tangent and normal of the first conic at the point P , bisect the angles formed by the two tangents in question." C. 33

258

ON THE DIAMETRAL PLANES OF A SURFACE OF THE SECOND ORDER.

In the case of surfaces, the equation of the circumscribing vertex as origin, is

I* f

2 7 _

*\ t* &

[39

cone referred to its

A _ (<** to

whence

A^pw + B = y>a? + a2c2 C=a2b2

H = -c!'a/3. Hence, omitting the factor & W + c2a2/32 + a2b2
and t h e equation of t h e system of diametral planes becomes a20y (c2 - ¥) a? + /327« (a 2 - c2) f + 7 2 a/9 (62 - a 2 ) ^

@= 0= +

ja {a2 (c2 - 68) + Z32 (6 s + c2 - 2a 2 ) -

+

a/3 {- a2 (c2 - 62) + /S2 (a2 - c2) +

+

ya. {a2 (a2 + P - 2c2) - /82 (a 2 - c2) + 7 2 (b2 - a2) + (a2 - c2) (62 - a2)} sctf

-

a/3 {a2 (c2 - ¥) - fi> (a2 - c2) - 7 2 (62 + c2 - 2a2) - (a 2 - c2) (c2 - 62)} y2^

-

/ 3 7 {- a2 (c2 + a2 - 262) + /32 (a 2 - c2) - 7 2 (62 - a 2 ) - (b2 - a 2 ) (a2 - c2)) z2x

-

7a {- a2 (c2 - b2) - /82 (a 2 + 62 - 2c2) + 7 2 (b2 - a2) - (c2 - 62) (63 - a8)} «2j/

2 7

2 7

(62 - a 2 ) + (62 - a2) (c2 - 52)] ,y^!

(c2 + a 2 - 2b2) + (c2 - &2) (a 2 - cs)} rf

+ {(a2 - b2) (b2 - c2) (c2 - a2) + (a4 + /3V) (b2 - c2) - (/34 +

2 2 7

a ) (c3 - a 2 ) - (7* + a2/32) (a 2 - V) +

_ C2) (2a 2 - 62 - c2) + /32 (c2 - a2) (26 2 - c2 - a 2 ) + 7 2 (a 2 - V) (2c2 - a2 - ¥)} xyz ; and since this is a function of a2 — ft2, b2 — c2, and c2 — a2, t h e equation is t h e same for all confocal ellipsoids; whence t h e known theorem, " T h e axes of t h e circumscribing cone having its vertex in a given point P, are tangents to t h e curves of intersection of t h e three surfaces, confocal with t h e given surface, which pass through t h e point P."

40]

259

40. ON THE THEORY OF INVOLUTION IN GEOMETRY. [From the Cambridge and Dublin Mathematical Journal, vol. n. (1847), pp. 52—61.] three conies have the same points of intersection, any transversal intersects the system in six points, which are said to be in involution. It appears natural to apply the term to the conies themselves; and then it is easy to generalize the notion of involution so as to apply it to functions of any number of variables. Thus, if U, V... be homogeneous functions of the same order of any number of variables x, y ..., a function ©, which is a linear function of U, V..., is said to be in involution with these functions. More generally © may be said to be in involution with any system of factors of these functions: or if U, V ... be given functions of a?, y, z..., homogeneous of the degrees m, n ..., and u, v, ... arbitrary homogeneous functions of the degrees r — m, r — n...; then, if © = ulI+vV+..., © is a function of the degree r, which is in involution with U, V... . The question which immediately arises, is to find the degree of generality of ©, or the number of arbitrary constants which it contains. And this is a question which, from the variety and interest of its geometrical interpretations, has very frequently been treated of by geometers, though never, I believe, in 'quite so general a form, (the number r has almost always had particular values given to it, except in a short paper of my own, on the particular case of two curves, in the Journal, vol. in. p. 211 [5]).1 There is also an analytical application of WHEN

1 The first suggestion of the problem is contained in a memoir of Euler's—" Sur une contradiction apparente dans la doctrine des lignes courbes," M6m. de Berlin, t. iv. [1748] p. 219. It is noticed also in Cramer's Introduction a Vanalyse des lignes courbes [1750]. The following memoirs also have been published on the subject: Plueker, " Eecherches sur les courbes algebriques de tous les degres," Gerg. Ann. t. xix. [1828—29] p. 9 7 ; " Eecherches sur les surfaces algebriques de tous les degres," p. 129; (a great number of memoirs on particular applications of the theory are contained in Gergonne;) Jacobi, "De relationibus quas locum habere debent inter puncta intersectionis duarum curvarum vel trium superfieierum dati ordinis, simul cum enodatione paradoxi algebraici," Crelle, t. xv. [1836]; Plueker, "Theoremes generaux concernant les equations d'un degre quelconque entre un nombre quelconque d'inconnues," Crelle, t. xvi. [1837], (but this last must be read with caution, as several of the theorems are incorrect, or at least stated without the proper limitations); and the Einleitende Betrachtungen, in Pliicker's "Theorie der algebraischen Curven" [1839]. The following memoirs of Hesse, containing developments relative to the case of three surfaces of the second order, may likewise be mentioned, "De curvis et superficiebus secundi gradus," Crelle, t. xx. [1840] p. 285; and " Ueber die lineare Construction des achten Schnitt-punctes dreier Oberflachen zweiter Ordnung, wenn sieben Sehnitt-puncte derselben gegeben sind," Crelle, t. xxvi. [1843] p. 147.

33—2

260

ON THE THEORY OF INVOLUTION IN GEOMETRY.

[40

the theory, of considerable interest, to the problem of elimination between any number of equations containing the same number of variables. Suppose, for instance, two equations, U = 0, F = 0 , when U, V are homogeneous functions of x, y of the degrees m, n respectively. To eliminate the variables it is sufficient to multiply the first equation by xn~l, xn~*y..., y™-1, and the second by a;™"1..., t/™""1, and from the equations so obtained to eliminate linearly the (m + n) quantities a;m+n~"1, xm+n~2y..., ym+'^-\ But in the case of a greater number of equations it is not at first obvious how many new equations should be obtained; and when a number apparently sufficiently great have been found, it may happen that the equations so obtained are not independent, and that the elimination cannot be performed. But in showing the connexion that exists between these different equations, the theory of involution explains in what manner a system is to be formed, which includes all the really independent equations, and gives the means of detecting the extraneous factors which appear in the result of the linear elimination of the different terms; but I do not see at present any mode of obtaining the final result at once in its reduced form free from any extraneous factors. Let X, Y, ... be given homogeneous functions of the same degree of any number of variables, and suppose a, /3... being constants, and the number of terms in the series being g; © contains therefore g arbitrary constants. If however, by giving to a, /S ... particular values alt /3X ..., or a2, & ..., and representing by ®t, ©2 ... the corresponding values of ©, we have identically ©i = 0, @2 = 0, ... (h equations); then the constants in © group themselves together into a smaller number g — h of arbitrary constants. This supposes, however, that the last mentioned equations are linearly independent; if there are a certain number k of equations (where <S>1( <E>2, ... are linear functions of ®u ©2...) which are identically satisfied, independently of the h equations, then the equations in question are equivalent to h — k equations, and the function © contains g — (h — k) or g — h + k, arbitrary constants. Similarly if the functions <& are not independent; so that the number of arbitrary constants really contained in © is always Consider now the case of a function ©, homogeneous of the r th degree in the variables m, y...{(0+ 1) in number}. Let V, V... be functions of the degrees m, n..., and suppose

® = uU+vV+ ... where u, v ... are arbitrary functions of the degrees i— m, r — n, ... {r is supposed throughout greater than m, n ...}. Suppose for shortness that the number of terms in the complete function of 6 variables, and of the order p, Le. the quotient ^ ^^e

, is

40]

ON THE THEORY OF INVOLUTION IN GEOMETRY.

261

represented by [p, ff] ; then the function © contains apparently a number ([r-m,

0] + [r-n,

6] + ...)

of arbitrary constants. But since we should have identically © = 0 by assuming u = LV, v=-LU, w = 0, &c... (L the general function of the order r — m — n), or u — MW, v = 0, w = —MU (M the general function of the order r — m—p) &c, the number j \ r must be diminished by

[r-m—n, ff\ + [r-m-p,

6] + [r-n—p,

&] + ...;

but the equations just obtained are themselves not linearly independent, and in consequence of this the number of arbitrary constants has to be increased by [r — m — n — p, 0]+ ... ; and so on.

Hence finally the whole number of arbitrary constants in the function 0 is N=\r-m,

0] + [r - n, 6] + [r -p,

6] + ...

— [i—m — n, ff] — [r — m—p, 0] — [r — n—p, ff] — ... + [r-m-n-p,

0]+ ... ± &c. &c

(A).

This however supposes that all the numbers r — m, 1—«..., r — m — n..., are positive: whenever this is not the case for any one of them, the corresponding term is obviously to be omitted. With this convention the equation (A) gives always the correct number of arbitrary constants in ©: it will be convenient to represent it in the abbreviated form

N={r : m, n, p, ... : 0}. An expression analogous to this, for the particular case of r=m, but incorrect on account of the omission of all the terms after the second line, has been given by M. Pliicker (Crelle, torn. xvi. p. 55), and even some of his particular formulas are incorrect. But proceeding to examine some particular cases: if r >m + n+p +... — 0 — 1, then in the expression (A) either no terms are to be omitted, or else the terms to be omitted reduce themselves to zero, so that N is given by this formula continued to its last term. It will be subsequently shown that in this case [r : m, n, p ... : 0} = [r, 0]— mnp ... ; or in the case of two or three variables, we have the theorem, "If a curve or surface of the order r be determined to pass through the ran points of intersection of two curves of the orders m and n, or the mnp points of intersection of three surfaces of the orders m, n, p\ then if r>m + n — 3, or r>m + n +p — 4, the curve or surface contains precisely the same number of arbitrary constants as if the mn or mnp points were perfectly arbitrary." This is natural enough; the peculiarity is in the case where r^>m + n — S, or m + n+p — 4. For instance, for two curves, r ^> m + n — 3, we have [r : m, n : 2}= [r-m,

2] + [r — n, 2] = [r, 2 ] - m n + [r-m-n,

2],

262

ON THE THEORY OF INVOLUTION IN GEOMETEY.

[40

or the new curve contains ^ [m + n — r — I] 2 more arbitrary constants than it would do if the mn points, through which it was made to pass, had been perfectly arbitrary; a result given before in the Journal, [5]. In the case of surfaces, if r $> m + n+p — 4. Then assuming r>m + n — 4, m+p — 4, or n+p — 4, we have {r : m, n, p : 3} = [ r - m , 3] + [r-w, 3] + [r-_p, 3] — [r — m — n, $]—[r — m—p, 3] — [i—n—p, 3]

= [r, 3] — m»|) — [r — m — « — £>, 3] ; or the surface contains ^ [m + n+p — r — I] 3 more arbitrary constants than it would do if the mnp points, through which it was made to pass, had been perfectly arbitrary. Similarly, in the case where r is not greater than one or more of the quantities m + n — 4, m+p — 4i, n+p — 4>. Thus in particular, if r be not greater than the least of these quantities {r : in, n, p : 3} = [r, 3] — mnp + [r — n — p, 3] + [r — rn—p, 3] + [r — m — n, 3] — [r — m — n —p, 3] ;

or the surface contains % [m + n + p — r — I]3 — ^[n + p — r — I]3 — ^ [m + p — r — l]s — ^[m + n — r — I]"

more arbitrary constants than it would otherwise have done. Again, for a surface of the rth order, subjected to pass through the curve of intersection of two surfaces of the orders m, n, [r : m, n, 3} = [r — m, 3] + [r — n, 3] — [r — m—n, 3] ; in which the last term, or \\m + n — r — I]3, is to be omitted when r^m + n — 4. The function of the rth order, which is satisfied by the systems of values which satisfy the equations of the orders m, n... contains, we have seen, [r, m, n, p...ff\ arbitrary constants; hence it may be determined so as to pass through this number, diminished by unity, of arbitrary points. But the equation being determined in general by the condition of being satisfied by [r, d] — l systems of variables; it will be completely determined if, in addition to the above number of arbitrary systems, we suppose it to be satisfied by a number N—[r, 0] — {r; m, n, p... : 6} of systems satisfying the equations above. Hence the theorem " The equation of the rth order which is satisfied by a number iV= [r, 0] — {r ; m, n, p... : 6}

of systems satisfying the equations of the orders m, n, p... is satisfied by any systems whatever which satisfy these equations." In particular—"The surface of the rth order which passes through a number [r, 6] - {r : m, n : 6}

40]

ON THE THEORY OF INVOLUTION IN GEOMETRY.

263

of points in the curve of intersection of two surfaces of the orders m, n,—or through [r, 6~\ — {r : m, n, p : 6} of the mnp points of intersection of three surfaces of the orders m, n, p,—passes through the curve of intersection, or through the mnp points of intersection." Thus a surface of the second order which passes through eight points of the curve of intersection of two surfaces of the second order passes through this curve; and any surface of the second order which passes through seven of the points of intersection of three surfaces of the second order passes through the eighth point. (The first theorem obviously fails if the eight points have the relation in question, i.e. if they are the eight points of intersection of three surfaces of the second order.) . Again—" The curve of the rth order which passes through [r, ff] — the points of intersection of two curves of the orders mn, passes through points of intersection." e.g. "Any curve of the third order which passes of the points of intersection of two curves of the third order, passes the ninth point."

{r : m, n : 6} of the remaining through eight also through

Consider next the following question, which [as regards particular cases] has been treated of by Jacobi in the memoir already quoted (Grelle, torn. xv.). " To find the number of relations which must exist between K(0 + 1) variables, forming K systems, each of which satisfies simultaneously equations of the orders m, n, p... respectively; the number $ of these equations being anything less than 0 j or being equal to 6, provided at the same time K— mnp ...." Suppose m (K — JV") relations in all. Similarly, if the equations of the orders p ... were given, JV' systems of variables might be assumed satisfying these equations, but otherwise arbitrary; the remaining JV — JV' systems satisfy ( — 2) (N—N') ... &c, which becomes negative, must be omitted. I t is obvious that we may write more simply JV = [ m , 6]-l-{m; JV'=[>, 0]-I-{n

n, p...6}, ;p...9}, &c

264

ON THE THEOBY OF INVOLUTION IN GEOMETRY.

[40

In particular, to find the relations which must exist between the coordinates of m« points in order that they may be the points of intersection of two curves of the orders m, n respectively: here K=mn, N= § [TO + 2]2 — £ [TO — n + 2]2 = \ (2mn — n? + 3ft), Xf' = ^ (n2 + 8n + 2), so that N —N' = m(m — n) — 1 which becomes negative when m = n; hence in general the required number of conditions is mn — 3M + 1 , but when m = n, the number in question becomes (n — 1) (n — 2). Passing to the case of surfaces; to determine the number of relations which must exist between the coordinates of mnp points, in order that they may be the points of intersection of surfaces of the orders m, n, p respectively. The number required is 3 {mnp-N)+2(N-2T')

+ (N'-N"),

where N=[m, 3]-l-[m-«, S]-[m-p, $\+[m-n-p, 3] (this last term to be omitted when in < n +p — 3), JV=[n, g]-l-[n^p, 3], If, for instance, m>n+p— 3, so as to retain the term [m — n—p, 3], and n>p, so as to retain the term JV* — N", the number becomes, after all reductions, 2mnp + np2 - 4>np ~2p2~l(p-1) (p -2)(p- 3), a formula given by Jacobi. If, however, n= p, this number must be augmented by unity. Again, for m < n +p — 3, the required number is 2mnp + np2 — 4«p — 2pi — \(p — 1) (p — 2){p — 3) — I (n+p — m — 1) (n+p — m— 2) (n+p — m — 3),

which however must be augmented by unity if m=w or n=p, and by 3 if m = n=p. But without entering into further details about this part of the subject, which has been sufficiently illustrated by the examples that have been given, I pass on to notice the application of the above theory to the problem of elimination. Imagine (6 + 1) equations between the (6 +1) variables, the first sides of these being, as before, rational and integral homogeneous functions of the variables of the orders m, n, p ... respectively. Writing m + n+p ... —6 — r, and multiplying the first equation by all the terms of the form xay&... of the degree r — m, the second equation by all the terms of the same form, of the degree r — n, and so on, there result a certain number of equations, containing all the terms af-y&... of the degree r. But these equations are not independent; and the reasoning in the former part of the present paper shows that the number of independent equations is given by the symbol {r : m, n, p... : 8}; the number of terms otf-yP... is evidently [r, 0]; and it will be shown immediately that for the actual value of r, [r, 0]-{r;

m, n,p...:0\=O

(B);

so that the number of quantities to be linearly eliminated is precisely equal to the number of equations, or the elimination is always possible. I may mention also that,

40]

ON THE THEORY OF INVOLUTION IN GEOMETRY.

265

supposing the coefficients of all the equations to be of the order unity, the order of the result, free from extraneous factors, may be shown to be [r-m, ff] + ...-2{[r-m-n,

6] + ...} +3 {[r-m-n-p, = mn... + mp... +np... +&c

6]+ ...}-&zc. (C),

(the equality of which will be presently proved) a result which agrees with that deduced from the theory of symmetrical functions; but I am not in possession of any mode of directly obtaining the final result in this its most simplified form. My method, which it is not necessary to explain here more particularly, leads me to the formation of a set of functions P, Q, X, Y, Z, 0 in number, such that Z divides T, this quotient divides X, and so on until we have a certain quotient which divides P, and this quotient equated to zero is the result of the elimination freed from extraneous factors. It only remains to demonstrate the formulae (A), (B), and (C). Suppose in general that (k) denotes the sum of all the terms of the form manb..., which can be formed with a given combination of k letters out of the (j) letters m, n, p •••', and let 2 (k) denote the sum of all the series (k) obtained by taking all the possible different combinations of k letters. It is evident that 2 (k) is a multiple of ($), {() denoting of course the sum of all the terms manb..., m, n... being any letters whatever out of the series m, n, p...}. Let g be the number of exponents a, b, ..., then (<£) contains [cf>¥ terms, also (k) contains [k]& terms, and the number of terms such as (k) in the sum 2 (k) is [<£]*~* +[ — &]*~*. Hence evidently

or, what comes to the same thing,

Let A be an indeterminate coefficient, cr a summatory sign referring to different systems of exponents; then

or, giving to k the values 1, 2 ... 0, multiplying each equation by an arbitrary coefficient, and adding, putting also for shortness crA (