206 CHAPTER VIIL 



According as I = or A = A', the number of sets of solutions of 



is Pi=2 n + 1 1 or P{f2 = l. We find by simple induction, 



P$ - 1 - (2- - 1) (2<- - 1 ) + 1), P#? - 1 == (2- + 1) (2< *> - 1). 



The number of distinct linear functions f is 2 /z(2m ~~ 2 ). In fact, 

 198) determines /J u in terms of /5^, $1, (j = 2, . . ., m), so that the 

 latter may be chosen arbitrarily in the 6r.F[2 n ]. 



It follows therefore, from 202, that 



aS.- (*- i)2 2 <>- <?-!, (m > i). 



By 203, we have the initial values 



We now readily obtain the formulae 



- 2 )... (2 2w l)2 2ra . 

 - 2 )... (2 2re l)2 2w . 



205. Theorem. - - T/wse substitutions of ft w/wc/fc satisfy the 

 further relation 



209) J(,ft y ,d) 



a subgroup of index 2 which any M { . extends to ft. If m> 2, 

 *s subgroup is identical with the group generated as follows: 



Nij lX ] (i, j = 1, . . ., m; H arbitrary in field). 

 If m = 2, ^ ^s identical with the group 



JI={MIM^ jy 8l M, 5i,x, ft, 1,1}- 



We first prove that every substitution of Ji satisfies 209). To 

 do this, it suffices to show that, if X be any substitution of ft 

 which satisfies 209), the products MiMjI., $-,/, x will also satisfy 209), 

 the case m = 2, being treated later. Let Z have the form 192). 



a) If the product -M^-Z be expressed in the form 



210) K 



we have J=1 



(7 _ _ "I AJ7 



fc=i''. !! 5 



