194 MA THEM A TICS: 0. VEBLEN Proc. N. A. S. 
Differentiating once more and repeating the process just described, we 
obtain 
f = 0, ^ (4.4) 
and in general, 
'dy^by . . x>y 
«^ y-y^,,,/ = 0. (4.m) 
5. Since the directions at the origin are entirely arbitrary it follows 
from (4.1) that 
(C,)o = 0. . (5.2) 
If we substitute / = in (4.3), divide by s^, and evaluate for / = 0 
we have 
r r r = o. 
Rewriting this with a,^,y permuted cyclically and adding, we obtain 
Since this form is symmetric and the ^'s are arbitrary, 
By a similar argument we obtain from (4.4) the relation : 
( 
b^cU , ^'cU , ^'cj, b^cU b'cu ■ b^cU \ .34. 
dy-'b/ by^by' by^by-" by'^b/ by'^by'^ by^^by^o 
By repeating this process we find that the sum of the {m -\- 2) {m -\- l)/2 
derivatives of the m^^ order of the functions, C, in which any set of m -\- 2 
integers, a,^,. . (Sm) appear as the subscripts of the Cs and the super- 
scripts of the y's, is zero at the origin of normal coordinates. 
6. We now define a sequence of systems of functions of {x^,x'^,. . .,x^), 
-^L/3...| > by the condition that the value of .^at any point {x^,%'^, 
(b^C^ \ 
I determined in the 
by\..by^), 
system of normal coordinates having {x^,x'^, . . . ,x^) as origin. 
From (1.2) it follows that 
^L/37...€ = A^y...^' (6.1) 
