Vol,. 8, 1922 
MATHEMATICS: 0. VEBLEN 
197 
polynomial in A's and B's with fewer than (m — 1) subscripts. By using 
(10.2) as a recursion formula this last polynomial is converted into a 
polynomial in the B's with fewer than (m — 1) subscripts. 
This completes the proof that each A is expressible as a polynomial in 
the 5's and determines explicitly the coefficients of the linear expression 
in the B's with m subscripts which forms a part of this polynomial. 
Each of the terms represented by the three dots in (9.2) is evidently 
a sum of products of A's and B's (such, for example, as A'^^j Bi^^^) of such 
a form that it must represent a tensor if the A's and 5's which enter into 
it represent tensors. The same remark follows directly for the terms 
represented by the three dots in (10.2). Hence by the use of (10.2) as 
a recursion formula it follows that the A's are all tensors. 
11. The rule for determining the permutations of the subscripts of the 
M — \ functions B which appear linearly in (10.2) can be regarded as a 
rule for tracing out the points and lines of a configuration analogous to 
the Desargues Configuration (cf . Veblen and Young, Projective Geometry, 
Vol. I, Chap. 2). To see this it is only necessary to observe that the M 
permutations of the subscripts of . ^ which give functions which 
are not identical according to (6.1) and (6.2) are in (1 — 1) correspon- 
dence with the points of the configuration obtained by taking a plane 
section of a complete m-point in a projective 3 -space. 
12. For the case m = 3 the formula (10.2) reduces to (8.4). For the 
case w = 4 the terms represented by the three dots in (9.2) are all zero 
and hence they are all zero in (10.2). Hence the latter formula reduces to 
= - {^B^otyd + 4 5^j8a5 + 3 B^^tb^ -f 2 B\^^^ + ^S/Say) (H-l) 
6 
Another formula which follows at once from (9.1) or (11.1) is 
From (9.1) there also follows at once the identity of Bianchi, 
Bapys + B^as^y -+- B^aySfi = 0, (11-3) 
as well as 
which is one of a sequence of important identities. 
