236 
MATHEMATICS: L. P. EISENHART 
Proc. N. a. S. 
where <p^^ is the covariant derivative of In these formulas there are 
no summations. 
5. By definition the contracted tensor is given by 
Rir=^Brj. (5.1) 
a 
Hence from the above formulas we obtain 
R'ij = Rij + n<pij - (pji- {n - 1) <Pi (pj 
i?',, = i?,,+ (n-l) {<pu-<Pi'). ^^'^^ 
Since B]ki = — B)ik, a skew symmetric tensor of the second order is 
defined by 
5y = SS^y. (5.3) 
a 
Now we have 
5',;;- = 5,,-+(w + l)(<^,, -^,,.). (5.4) 
From (5.1), (4.1) and (5.3) we have 
Hence 5^ = 0 is a necessary and sufficient condition that Rij be sym- 
metric. This result was established by Veblen in a paper presented to the 
Academy, April 25, 1922. 
Veblen (these Proceedings, July, 1922) has shown that the identity 
of Bianchi, namely 
Bjkim + Bji^k + Bjtfiki = 0 (5.5) 
holds in the geometry of paths; an equivalent result was announced by 
Schouten (Jahr. Deut. Math. Ver., vol. 30, 1921, Abt., p. 76). In con- 
sequence of (5.5) we have from (5.3) 
Sijk + Sjki + Skij 0, (5.6) 
or in other form 
-f ^ + = 0. (5.7) 
bx" bx' bx^ 
We have shown {Bull. Amer. Math. Soc, 1922) that (5.7) is a necessary 
and sufficient condition that 5^- is expressible in the form 
(5.8) 
' ^x' bx' 
where is a covariant vector, any one component being arbitrary and the 
others being determined by quadratures. As an immediate consequence 
we have the theorem : 
If Sn is any geometry of paths, spaces 5'„ can be found whose paths are in 
one-to-one correspondence with the paths of S„ and such that the tensor R'ij is 
symmetric ; the determination of involves an arbitrary function of the x's. 
6. For a Riemann space the tensor Rij is symmetric. Consequently 
