Vol. 8, 1922 MATHEMATICS: L. P. EISENHART 
235 
it follows that s' must be the same for all paths just as 5 is in (2.1). From 
(3.7) it is seen that for this to be the case we must have 
v/-f = 4'{.s), (3.8) 
as 
xf/ being the same function for all the paths. Differentiating and making 
use of (2.1), we obtain 
^a^—- ■— = 4^\s), (3.9) 
as as 
where are the components of the covariant derivative of the vector 
as defined in the former paper, that is 
<Pa^ =^^J - <PiTU' (3.10) 
Consider first the case when i^ in (3.8) is constant, that is when (2.1) 
admits a linear first integral. From (3.9) it follows that 5„ must be such 
that there exists a vector so that ip^^ -f <^^„ = 0. When this con- 
dition is satisfied by one, or more vectors, the equations of the paths can 
be written in two, or more ways, in the form (2.1). Spaces which admit a 
family of parallel covariant vectors, considered by the author in a former 
paper (these Proceedings, July, 1922) are of this kind. 
lixj/' = const, in (3.9), equations (2.1) would have to admit a quadratic 
first integral. Another case of this sort arises when t//' = ai//^ + 6, where 
a and h are constants. Although these special cases are interesting and 
lead to types of spaces for which the equations (2.1) are not unique, it is 
clear that this is not possible for a general 5„. Similar results hold when 
is not constant. For on the elimination of s from \p and xp' in (3.8) 
and (3.9) we should obtain a relation between the first derivatives of the 
x's, which could not hold for all the paths. Hence: 
For a general space the equations of the paths can he written in the form 
{2.1) in only one way, the parameter s being such that it may be replaced by as 
+ b, where a and b are arbitrary constants. 
4. The components of the curvature tensor in 5„ are defined by 
■^^jki ^ ~^ ^5 ^^cck —^Jk (4.1) 
ox ox 
If we indicate by primes the functions for 5„', we have from (3.4) and 
(3.6) 
B%i = B%i a, i, k, I 
B'ijk = B'ijk + <pij — ifjk {i, j, k 
B'U = B%i + fj, -vj >Pk ii, ./, k 4^), (4.2) 
B'iji = B'iji — ifji + 2 ipij — ifi ifj (i, y 4=), 
