230 CHAPTER X. 



In order to express more compactly the canonical form S f we let 

 a, b, c denote an arbitrary one of the respective sets of integers 



a) 1, A + 1, a x + a 2 + 1, . . ., % + a 2 -\ ----- h a r + 1; 



b) 1, & A + 1, Z^ + fca + l,..., ^ + 65,+ +&, + !; 



c) 1, q+1, q + c^ + l,..., Ci + c a H ----- he, + 1. 



Also let A denote any integer ^ a not an a, J5 any integer <^ ft 

 not a &, C any integer < y not a c. The canonical form 8j_ may 

 now be written as follows: 



-l (i = 0, 1, . . ., Jc 1) 



I (i = 0, 1, . . ., Z - 1) 

 -i (i = 0, 1, . ., q - 1). 



An arbitrary linear homogeneous substitution l l on these indices 

 replaces ^- by a linear function 



233) ZDi^x + Z JEi'i &. + Z ^;i ^ w , 



where (as henceforth) the summation indices have the series of values 



*-0,l,...,*-l5 A = 0, 1,...,Z-1; ^ = 0,1,...^-!; 

 w= !,...,; t; = 1, . . ., |3; w = 1, . . ., y. 



Jw onfer ^fea< T x &e commutative with $ x ^ is necessary tliat 233) 

 involve only the indices rj iu (u = 1, ...,). Equating the functions by 

 which jTjjS^ and /S^ J x replace ^,- a , we get 



Equating the coefficients of the ^'s and 's in this identity, we get 



i^ LI BIB i + LI BIS* 



Since Jf f =(= i^, the third equation gives Ell\ = 0, where & is 

 any integer > 1 of the set b). If 6 1 is a J5 ; the fourth equation 

 gives JE3?_a = 0. In the contrary case, & 2=)=5 1, and the third 

 equation gives EH2= 0. Similarly, according as & 2 is or is not 

 a J5, the fourth or third equation gives J3&_ 3 = 0. Proceeding in 

 this manner, we find that every JEJ J = (A = 0, ...,? 1 ; v = 1, . . ., /3). 



