SYMBOLIC NOTATION 
0£l9»72 
d&dvi 
Subtracting, we get 
Hence the lemma is true when r = 1. It now follows at once 
by induction that 
where the first summation extends over all of the 
k(k — l) . . . {k-r-\-1) permutations s\, . . . , s T of 1, . . . , k 
taken r at a time, and the second summation extends over 
all of the 1(1— 1) . . . 1) permutations h, . . . , t T of 
1, . . . , l taken r at a time. 
Corollary. The terms of (l) coincide in sets of rl and 
the number of formally distinct terms is 
(k—r)\ (il—r)\ rl \r)\r / 
For, we obtain the same product of determinantal factors 
if we rearrange $i, . . . , s r and make the same rearrangement 
of . . . , t T . 
45. Proof of the Fundamental Theorem in § 42. Let K be 
a homogeneous covariant of order co and index X of the binary 
form / in § 40. By § 40, the general linear transformation 
replaces /=a x n by