Determination of Some Properly Irregular Cyclotomic Fields

MA THEMA TICS: M. S. KNEBELMAN

156

PRoc. N. A. S.

where p = 0, 1, 2, ..., i = 1, 2. 8. An immediate proof of the theorem that the second term of the disturbing function contributes nothing to its constant part is furnished by the last theorem of § 5. In the usual notation, R =m _mrS A2 = r2 + r'2-2rr'S,

S = COS2 1/2i COS(+ C-f'-C') + sin2 1/2i COS (f + o + f' + '-2). On the multiplication of R by r2r'2 and the substitution of a(1 - e2)(1 + e cos f)'- for r in r3S we see at once that when r3S is expressed as a sum of periodic terms, every term must contain f' in its angle and there can therefore be no constant term. Proc. Nat. Acad. Sci., 16, 1(1930).

CONTENT-PRESERVING TRANSFORMA TIONS BY M. S. KNUB3LMAN DXPARTMT Olt MATnEMATICS, PRINCETON UNIVERSITY

Communicated December 27, 1929 By a generalized metric space of n dimensions we shall understand one in which arc length is defined by a positively homogeneous function of the second degree in dxl, ..., dx"; xl, ..., xe being the coordinates of any point. In a space of this sort the fundamental tensor gij(x, dx) is homogeneous of degree zero in dx instead of being independent of dx, as it is in a Riemann space. For this reason all measurements in our space are relative to a chosen direction. If V(,)(x, dx), a = 1, ..., k . n, are k independent contravariant vectors, that is, if || ) is of rank k, the content of the k-cell defined by them is given by

-I

k

V2

=

gal... a k; i..ik

a=l

(a) \(a)s

where the summation convention is used only for repeated indices and where ga,.. -i, . denote the k-rowed minors of the non-vanishing determinant gij the rows and columns of the minor being indicated by the a's and i's, respectively. Our problem is to find the conditions that the fundamental tensor must satisfy in order that there shall exist a finite continuous group of extended point transformations which shall carry every k-cell (for a given k) into a k-cell of the same content.

I I,

Vo.4 16, 1930

MA THEMA TICS: M. S. KNEBELMA N

157

If k = 1 the transformation in question-if it exists-is a motion of the space into itself and in that case the content of every k-cell for every k = 1, ..., n is preserved. As in the problem of motion we shall consider the infinitesimal transformations of the group, since the general theory of finite continuous groups establishes the existence of the corresponding finite transformations. Let (2) x = xk + (X)au be an infinitesimal transformation defined by the vector {ix. Then the components of the fundamental tensor at x are given by

gij(Xj dt)

=

gij(x, dx) +(Xk e +

gij

dxze

(3)

) au + .

while the vector X'(x,dx) is carried into the vector

Xi(x dx)

Xi + Uaxi X8u,

=

(4)

in accordance with the law of transformation of vectors. It therefore follows that

V2

=

[gai...ak; ii...ik +

+

(ga; i,m

ga;i.m

k

a-;

(ea6

*I .B(a) * a1

dx)5u]. +

aasu)(8at

+

t

taau),

where a subscript preceded by a comma denotes partial differentiation with respect to x and that by a period, with respect to dx. If a content-preserving transformation is to exist in the given space V2 = V2 for all vectors, X(,) and the equations arising from making v2 identically equal to V2 will be the differential equations that the vector e must satisfy in order to define an infinitesimal content-preserving transformation. Neglecting power of bu higher than the first these conditions give k

(ga; i,mt"

+ ga;

k

i,

m

d)

aa

act (4a

+

o

a aia 541) ta

ni

a=

where ga'indicates the product of all factors except the ath one.

MA THEMA TICS: M. S. KNEBELMAN

158

PROC. N. A. S.

Because of symmetry these equations are reducible to

2(gr;j,m tm + gr;j.t. t.,k

dxp)

tk,

+ 2 gr. .a.

Ja + 'Jaa'a + 8j4~ {'7a + 0J. = 0 '

i.3.. J .

(6)

Using the fact' that k

grl. m ji

.

k ...jr Ig7k ai1 r

A

k and multiplying (6) by II gr8Ji, contraction gives after considerable re8i2 duction, 2(gylj,,m gm + g7ljl.m tPM 671 72 Y

+2

k 2

51721 fa 7kgkj glaia (-aa (g7jiatm + g7aia,m t,m dxi) £171A ta -k

a-

+ g7jizli.

k

k

(5at

,

+

a; t2i

+ si

;

..k 71g,raja (5,aC + sba +~~ + 'Z 5717Y2 7Y a-2 r17'2. a.. 7k gj 7a a ja + ra 4 +a da ..

Since'l n"1z,7k= (n

~ 'a

0

supra

))I Si these last equations become

(n - k)hi, + (k - 1)gijh = 0,

(7)

where

h

gsa + g

+

e

Si

+ a

a

de

(8)

and h Zgihji. 0 are the generalized equations of lling, It has been shown2 that hi a solution of which-if it exists-defines a motion of the metric space into itself. Multiplying (7) by g"u, contracting the i's and j's we get k(n - 1)h = 0 and since k 0 O and n 0 1, h = 0. Hence (n-k)hij = O and if k < n, = 0 and the desired transformation must be a motion. We therefore have If a generalized metric space is such that there exists a transformation preserving the content of every k-cell for a given k < n, this transformation is a motion and thus wiU preserve the content of every k-cell for every value of k.

hij

159

MA THEMA TICS: M. S. KNEBELMAN

voj. 16, 1930

The only case that remains to be considered is k = n. The volume defined by n independent vectors is given by I V = Vg i(a) I and therefore

v=

+ [v (g

a

k + ad a m ) 6u] { \(a) I

+

a)

UX(8)

VWI bgX

= V + IX(a) I.*h. u

I

Hence if volume is to be preserved we must have h = 0, since |(a) does not vanish. Or explicitly written, the necessary and sufficient conditions that t' must satisfy in order to define a volume-preserving transformation is ar

t+ . aMIog vi + aogVg a

dxj

=

0.

(9)

The above equation must hold for all values of dx and is therefore equivalent to a system of partial differential equations which must admit a solution if the space is to admit a volume-preserving transformation. A solution of (9) may exist which does not satisfy hij = 0 and therefore a volume-preserving transformation need not preserve the content of cells of lower dimensionality. In fact, if the space be Riemannian, or more generally if the space be such that g is a point function, a volume-preserving transformation always exists; for in this case (9) becomes = 0. Therefore a vector density with vanishing divergence 6(evelaixi defines a volume-preserving transformation.

NoTi IN PROOF:

In an entirely similar manner it can be shown that every transformation which carries every k-cell, k < n, into a proportional k-cell also carries every vector into one of proportional length. That is, if a transformation is such that VI2 = Vka + kf(x) Vi.6u, then La = L2 + f(x)L'8u, L being the length of any vector. 1 For the theory of determinants and generalized Kronecker deltas used in this paper see CAMBRIDGE; TRAcT No. 24 by Oswald Veblen. 2 Cf. M. S. Knebelman, Amer. J. Math., 51, 1929 (527-564).