68o 
MATHEMATICS: L. P. EISENHART Proc. N. A. S. 
of a in this determinant divided by a. This assumption is equivalent to 
the equation 6^44 = 0. 
III. The surfaces V = const, form part of a triply orthogonal system 
in the space, Ss, of coordinates Xi, X2, Xz. 
IV. The orthogonal trajectories of y = const, in Sz are paths of the 
particle, in the sense that the codrdinates X\, ^2, ooz, of a world-line de- 
termine a path in Sz of a particle in the gravitational field for which the 
world-line is the representation in terms of space and time t. 
V. The form (5) is euclidean to a first approximation. 
2. In order to simplify the equations, we take (5) in the form 
1.2,3 
dso'' ^^^aidxi\ (8) 
» 
If we choose 5 for the parameter along a world-line and apply the Euler 
equations of condition that the integral 
be stationary along the line, we get 
+ ^ ^loga, d_Xi dxj _ doj /dxjV y^_y(^Y = 0 (9) 
ds"^ j bxj ds ds 2ai j dxi \ds I dx,- \ds) 
I - T- 
where ^ is a constant. In these equations and hereafter the symbol S 
j 
means the sum for / = i, 2, 3. 
We inquire under what conditions the path of a particle in is a 
geodesic, that is a curve along which the integral X^aidx^ is stationary. 
i 
When So is taken for the parameter along such a geodesic, we find that 
dHj _^ d log aj dxj dxj _ j_ bo^- (dxA^ _ ^ ^^^^ 
dso^ j dxj dso dso 2ai j c>Xi \dso/ 
From (4) and (10) it follows that the parameters 5 and So along a world- 
line and the corresponding path in Ss are in the relation 
ds 
o = - 1 ds. (12) 
With the aid of this relation we find that in order that equations (9) 
and (11) define the same curves in S3, we must have 
dVdx, _lbV^Q (i= 1,2,3). (13) 
dso dso ai bxi 
These are the conditions that the path be an orthogonal trajectory of 
the surfaces V = const. When we require that equations (11) and (13) 
