204 
MATHEMATICS: J. L. SYNGE 
Proc. N. a. S. 
PRINCIPAL DIRECTIONS IN THE EINSTEIN SOLAR FIELD 
By J. L. Syngk 
Department of Mathematics, University op Toronto 
Communicated, May 23, 1922 
In the Newtonian manifold of space-time there is at any point one prin- 
cipal direction, namely that for which the three space coordinates are 
stationary. In the hyperbolic space-time of the vSpecial Relativity Theory, 
there are no principal directions, the only directions intrinsically definable 
being those forming the cone 
- dx^ - dy^ - dz" + = 0. 
In the previous paper {Proc. N. A. S., p. 198) several types of principal di- 
rections have been defined for a Riemannian A/'-space, but the field equa- 
tions 
render indeterminate the principal directions of Types I, II and III as 
there defined. Kisenhart {Proc. N. A. 5., Vol. 8, No. 2, p. 24) has shown 
that those of Type III are indeterminate for all three forms of E^instein 
space free from matter. However, it would appear that those of Type 
IV might exist in the Solar Field; they correspond to stationary values 
of the invariant function of direction 
a „sih „s2t2 „s3t3 n 
^ — g g g ^SlS2,S3S^ht2,ht — -T' 
as as 
But, as will be seen, the value of 6 at any point proves to be independent 
of direction and therefore the principal directions of Type IV are inde- 
terminate. However, since 6 is an invariant function of position varying 
from point to point, there will exist principal directions corresponding to 
stationary values of d^d/ds'^ for geodesies drawn in all possible directions. 
These principal directions are given by 
r_^-i^n^ld., = fe&, (. = 1.2,3,4) 
(1) 
and are a generalization of Type II. 
The manifold under consideration is of four dimensions, and, in accord- 
ance with the conventions employed in the foregoing paper small 
Roman indices imply a range or summation from 1 to 4, small Greek 
indices from 1 to 3. The line element is given by 
2 _ dXjfi dXfi 
where 
gn = — {l-k/xi}-'^, g22 = -xj, g33 = - xisin'^X2, 
gu = 1-k/xi, gmn = 0 {m^ n). 
