234 
MATHEMATICS: L. P. EISENHART 
Proc. N. a. S. 
We say that corresponding points in 5„ and S'„ are those for which the 
coordinates x have the same values in the two spaces. If a path P in 5„ 
is to correspond to a path in 5'„, there must be a relation of the form 
s = J{s') connecting the parameters 5 and s' at corresponding points. In 
order that the functions of s' shall satisfy (2.1) and (3.1) we must have 
r2 dx dor 
ds ds 
From these expressions it follows that we must hav^e 
[fv- r V^' (v^^ \dx'ldx- dx' 
0 = 1,..., ^). (3.3) 
If it is desired that all the paths of 5„ can be brought into one-to-one cor- 
respondence with those of 5'„, these equations must be satisfied identically 
dx 
by the quantities — , otherwise we should have differential equations 
ds 
of the first order satisfied by all the solutions of (2.1). The conditions 
that these equations vanish identically are 
V'\i = Vi T'% = Viu {t, /, k 4=), (3.4) 
and 
T'i-2Tlj = n-2Tij, T%-T% = ^,-4 {i, j, k 4=) (3.5) 
Equations (3.5) are equivalent to 
r1, = r;, + 2<pi, t% = Tij + <pj. (3.6) 
where the functions cpi (i = I, . . n) of the x's are defined by (3.6). By 
making use of equations (4.2) of the former paper which give the relations 
between the F's in one set of coordinates x and those for another set, we 
prove that the functions (pi in (3.6) are the components of a co variant 
vector. 
If we substitute the expressions for T'^^ from (3.4) and (3.6) in (3.2) we 
have 
^=-2.«^. (3.7) 
P ds 
If we substitute in the right-hand member the expressions for the x's as 
functions of 5 for a given path, we get the relation between 5 and s' in order 
that these functions shall satisfy (3.1) also. Consequently (3.4) and 
(3.6) are sufficient as well as necessary that the paths of 5„ and S'„ be 
in one-to-one correspondence. 
We are in a position now to determine under what conditions the paths 
of S„ are defined also by equations (3.1). From our definition of paths 
