LINEAR GROUP WITH QUADRATIC INVARIANT. 163 



But the theorem is true for a{, a[ y cc' s if true for the quantities 

 '/ = atf - aiy 9 < EE < a" = - 



where /3 2 - y 2 = 1. In fact, if 8" replaces & hy o^ + a>& + ' 3 %, 

 then 0?;/S" will replace ^ by &+&+&. The^-l sets 

 of solutions in the GF[p n ~\ of the equation |8 a y 2 =l are given by 



where r runs through the series of marks =f= of the field. Hence 

 ft + y may be given an arbitrary value x =j= in the field. The 

 theorem being evident if a[ = 0, we exclude this case. Then a" = 

 a[ (ft + y) may be made to assume an arbitrary value except zero, and 

 hence, if ^) w >3, a value for which 1 a" 2 is a square in the field 

 ( 64). For jp n = 3, a[, cc^, cc' 3 are each 1, so that we may 

 evidently take 



where C and 5" are products formed from C 19 C 2 , C 3 . But, if C be 

 the product of an odd number of the d, we note that 



^1 ^123 = ^2 ^3 ^123 ^1 ^2 ^3 



We may therefore assume that C and .BT are each products of an 

 even number of the d and therefore derived from the given generators. 



176. Suppose next that 1 is the square of a mark i of the 

 6r-F[p n ], while ft is -a, not -square. There exist p n -\-\ sets of solutions 

 in the field of the equation 



149) ' 0t + JL y _i. 



But the theorem is true for a ly &%, or 3 if true for 



Indeed, if 8' replaces & by & + i6 8 + *&, then 0%IS' will 

 replace g t by a^ + or 2 | 2 + cc s ^. 



There are at least ~-(jp" + 1) sets of solutions of 149) for which 

 the values of a' 2 are distinct; for, upon eliminating /3, we obtain a 

 quadratic for <y. But, by 67, there exist only y(p n 1) marks ig, 

 and hence as many distinct marks J, for which 



(^|) 2 + 1 = 1 - | 2 = not-square. 



11* 



