THE ABELIAN LINEAR GROUP. 95 



But S is to be commutative with every pair L^ and L'I^. It follows 

 that S reduces to the form 



By the first type of Abelian conditions given under 79), we have 

 aa= 1. Since S is not in K, the an are not all -f- 1 and not 

 all 1. Transforming S by a suitable product of the form Pi r P 2 ,, 

 we may suppose that a n = 1, ff 22 = 1 in S. Then J contains 

 Ni~2,(tSN'] L) 2 i f tf which replaces | t by ij 2 k o-^ 2 and is therefore (since 

 p =j= 2) not of the form S. Taking it in place of our initial sub- 

 stitution S, we are led to the case next considered. 



Suppose that not all of the above products reduce to the iden- 

 tity J; for example, let 



If S~ l replaces ?; t by the linear function CD/ A, the product denoted 

 by Si has the following form, in which the coefficients of il have 

 not been calculated: 



i;..-=g/-a,-iCD (* = 2, ..., m), 



From Si we proceed to determine a substitution =J= / belonging 

 to J and leaving 2m 3 indices unaltered. S 1 itself is such a sub- 

 stitution if KH== Pn= (* = 2, . . ., m). In the contrary case, the 

 transformed of S by a suitable P 2; or P^jM^ will have <* 21 =(= 0. 

 Consider therefore S t when a 21 =f=0, and introduce the new indices 



ii = if -^ is, fy- = ^ -^-?2 (* "^ 3, . . ., w), 



an operation equivalent to the transformation of S^ by the following 

 product T belonging to the group G: 



where 



We obtain the substitution 5 2 ^ T~ l SiT, leaving fixed 2m 3 in- 

 dices, viz., 



1- =(&- a fl co) - ( - or 21 o)) = I,-, ^ = ^ (i = 3, . . ., m) 



