196 
MATHEMATICS: 0. VEBLEN 
Proc. N. a. S. 
9. In order to extend the formulas of §8 to our sequence of sets of func- 
tions we observe that by repeated covariant differentiation of 
(8.1) we obtain 
BaPyS...^ - ^ ;5 f — ^ ^ + . . . (9.1) 
where the term on the left of the equality sign represents an (m — 3)rd 
covariant derivative of Bl^^^, and the three dots at the right represent 
terms involving derivatives of the F's of order less than m and covariant 
derivatives of B^^^^ of order less than m ~ 3. By writing (9.1) in normal 
coordinates so that the F's in the right member become C's and then 
evaluating at the origin by means of the definition of the A's, we obtain 
in which the three dots represent terms involving A's and B's with fewer 
than m — 1 subscripts. 
The equations (9.2) by themselves do not determine the A's uniquely 
in terms of the B's. When taken in connection with the equations in 
§ 6, however, they can be solved. 
10. In order to find this solution we observe that among the A's with 
a given set of n subscripts any one is determined by its first two subscripts, 
and these two subscripts are interchangeable (§ 6). We consider the 
following w (m — l)/2 permutations of the subscripts {^ay. . .^,) {y^a. . .^),. 
(7«5...^), (5t^...^), Wa...^), (dae....^), {e8y...O, (ey^...^).... 
The A's with any two successive permutations of this set as subscripts are 
capable of entering in an equation of the form (9.2). There is thus deter- 
mined a set of M — 1 equations like (9.2), where M = -m{m — 1). 
Let the first of these equations be multiplied by {M — 1), the second by 
(M — 2). . . and the last by 1. On adding the resulting equations we 
obtain 
(M-1) + (M-2) + (M-3) + • . . ^^^^^ 
The parenthesis of the right member is zero by (6.4). Hence (10.1) re- 
duces to 
AU. . = i I (M- 1) 4.,. . -f (M-2) ^+ +...| 
(10.2) 
in which the coefficients of the terms in the parenthesis are the integers 
from 1 to M — 1, the permutations of the subscripts of the 5's are those 
indicated in the paragraph above, and the final three dots represent a 
