222 CHAPTER X. 



Hence K must be a root of the characteristic equation 



CCooK. 



cc 2m 



Corresponding to each root K, we may determine at least one set 

 of solutions fa of the above linear equations and hence one invariant 

 function 77. 



If A(-ZT) = has m distinct roots K 17 K 2 , . . ., K m (not necessarily 

 in the initial GrF[p n ], we reach m linear functions %, %, . . ., tyw, 

 which S multiplies by) JS^, J5T 2 , . . ., K m respectively. These functions 

 are linearly independent with respect to the variables (;. For, if 

 constants exist such that 



Ml + ft % H ----- H 



= 0, 



we have on applying the substitutions S, S 2 , . . ., 



identities 



+ ^2^2% H ----- H -^m^ OT ^ m = 0, 

 + jqpa% + + JE^^EE 0, 



1 1 the further 



But the determinant 

 1 1 



.1 



f, j==l . . . 771 



Hence 



0. 



Introducing the linear functions ^ f as new indices in place of 

 the I,- the substitution /S takes the canonical form 



S': 



,..., m). 



If we take in place of % a suitable multiple of ^, we may suppose 

 the reduction of 5 to $' to be accomplished by a transformation of 

 indices of determinant unity. 



Suppose, however, that the roots -of A(JST) = are not all 

 distinct. Let 



