168 CHAPTER VII. 



where C and K denote products of an even number of the C{ [com- 

 pare the end of 175]. 



Suppose next that the above sums are not all zero, for example 1 ) 



J + J + 74*o. 



We have proven that, for every set of solutions of 

 152) tt s +/} *+.lj, 2= l, 



there exists a substitution Z of the group, 



which therefore satisfies the relation 152) and the following: 



/3"=0, etc. 



If there be a substitution S' in our group which replaces | x by 



m 1 



I. 



where 



then the group will contain Z$ f which replaces ^ by 0?^ The prop- 

 osition is therefore true for the quantities y if true for i, 2, #m> 

 cr 3 , 4 , . . ., a m _ i. We may thus make our proof by induction from 

 m 1 to m by showing that it is possible to choose a, /3, y among 

 the sets of solutions of 152) in such a way that i = 0. We may 

 suppose that a^ =|= fy since otherwise the proposition is already proven. 

 If | + f = 0, then * 8 4= 0. From J, = 1, it foUows that p 

 is a square, say ft = 1. Then the values 



satisfy 152) and make 1 = 0. 



If a\ + a| ^ ^> th e condition 152) combines with a'l = to give 

 a single condition for /3 and y: 



1) The treatment for a case like ? -f af -f- =|= o is quite similar, taking ft = 1. 



