Vol.. 6, 1920 
MATHEMATICS: L. P. EISEN HART 
be consistent, we find that it is necessary and sufficient that be a 
function of F, say 
Ay = v{V), (14) 
where AF is the first differential parameter of V with respect to the form 
ds'^o' F'or the form (5) its expression is 
When (14) is satisfied by a function F, the surfaces V = const, are said 
to be geodesically parallel. Hence we have the theorem: 
A necessary and sufficient condition that a path in a gravitational field 
he a geodesic in S3 is that it be one of the orthogonal trajectories of the surfaces 
V = const., which must form a geodesically parallel family. 
3. The function V is interpreted as the velocity of light in the field, 
and consequently along a world-line of a ray of light ds = 0, as follows 
from (4). In order to find the equations of these world-lines we apply 
the Fermat principle that J^dt be stationary along such a line, that is 
the integral i_ XaidXi\ This gives the equations 
djxi dxi y b_ a_^ dxj _v^yb (aA /^A ^ ^ Q 
dt^ dt ^ dxj V dt 2ai ^ bxi \FV \dt ) 
When we require that the path of a ray of light be a geodesic in Sz, we 
obtain equations (13) and (14) as formerly. Hence: 
When the orthogonal trajectories of the surfaces V = const, are paths of 
a particle in a gravitational field, they are also the paths of a ray of light; 
and conversely. 
4. In accordance with III we choose the coordinates Xi, X2, Xs, so that 
F is a function of Xi alone and we have (8). Then from IV, which is 
equivalent to (14), written AF = <p{xi), and II we have 
ai = , a2a3 = — , (16) 
where the prime indicates differentiation with respect to Xi and \p is inde- 
pendent of Xi. 
If dso^ = 'ZoidXi^ is the linear element of euclidean space, the functions 
ai must satisfy the six equations of Lame.^ If ai is to be a function of 
Xi alone and the second of (16) is to be satisfied, it can be shown that the 
coordinates Xi can be chosen, so that 
d~So^ = dxi' -f Xi'^iaW + tMxz^), (17) 
where a and r are independent of Xi and satisfy 
dx2 0 5^2) ~^ dxs (r dxs) ~^ ^ 
This is the condition that the expression in parenthesis in (17) is the linear 
element of the unit sphere.'^ 
