LINEAR GROUP WITH QUADRATIC INVARIANT. 195 



Let <? be a root of the equation 



which is irreducible in the GF\_p n '\ in virtue of the irreducibility 

 of q. The second root is therefore ff^^o 7 , so that <S~G = A 2 . The 

 substitution 



z- x-k-tffc, r-t,-% 



transforms the function F=XY-\- 2 9? 2 into /". Let a, ($, y, d be 

 any set of marks in the GrF[p 2n ~\ subject to the condition ad $y = 1. 

 Then F is absolutely invariant under the substitution [see 181)] 



U: X'=a 



If we regard 1 ) | 1; ^ 1? 2 , % to be arbitrary marks of the 6r.F[jp n ], 

 F will be conjugate to X with respect to the GF\_p*\, while F will 

 be absolutely invariant under the following substitution conjugate 

 to U [see 182)] 



If therefore the product Z7Z7 be expressed in terms of the indices 

 %i> %> fe; %^ ^ e ^suiting substitution TF will leave f absolutely 

 invariant and have its coefficients in the 6rF[^ n ]. To give the 

 explicit form of W 9 let U and U become C/i and U^ when written 

 in the indices g,-, ^,-. Since the reciprocal of Z is 



we find for Z7 t the substitution 



li % 2 



a d o$ Get ft y 



y GY a 



f} 0ft d 



The coefficients of U t are conjugate to the corresponding coefficients 

 of U v The product TF= C/i^ is readily found to be the substitution 



1) This interpretation is not a necessary one in view of the later explicit 

 calculations. 



13* 



