Vol,. 8, 1922 
MATHEMATICS: 0. VEBLEN 
195 
From the fact that all the subscripts of after the first two 
■correspond to differentiation of it follows that 
^a/37...i = ^^a5...7?> (6.2) 
where b. . .7] stands for any permutation of the subscripts, t . . . ^. From 
(5.3) there follows 
"f" ^^70; "1" ^7a(8 ~ 0' (6.3) 
This is a special case of an identity, 
AUy...^ + Ky^...i + ••• 0 (6.4) 
in which there are m (m — l)/2 terms, each pair of the nt subscripts 
a^y . . . ^ being the first pair in one and only one term. This identity 
follows directly from the last theorem of §5. 
7. The system of functions A^^^^ ^^ with m subscripts (m^ 3) is a 
tensor of order m -\- i. This theorem can be inferred directly from the 
invariantive character of the normal coordinates. But we prefer to 
prove it here by showing how to express the functions A explicitly in terms 
■of the curvature tensor and its covariant derivatives which have already 
been proved to be tensors in the paper by Professor Kisenhart and the 
writer. 
8. The curvature tensor is defined by the equation, 
(This is the negative of what we denoted by in the former paper.) 
If it is computed in a normal coordinate system and evaluated at the 
origin, P, of these coordinates, it must satisfy the equations. 
These equations are equivalent to 
By means of (5.3) and the definition of Al,gy this leads to 
Ai,, = \ (2Big, + S;,^). (8.4) 
By means of the well-known identity, 
B^aPy H~ "f B^ffya — 0> (8.5) 
this reduces to 
^a^y = - (B^a^y "f B^^^y) . (8-6) 
O 
The identity (8.5) is itself a direct consequence of (8.2). 
(8.3) 
