140 CHAPTER V. 



In the latter case, the substitution S thus obtained has (since p = 2) 



+'-!. 

 Transforming it by T=T^ r T^ t i., we obtain in I the substitution 



8'=-8-*-T 



where S 1 denotes the substitution 



Hence for S' m S^ T the coefficient of | t in |{ is 



Setting for brevity ag+ 1 =a, a mark =)= in the 6rF[p], we find, 

 since r^+^l, that 



aP'+i = 1 + a (r - l)(r^ - !)( - ^ - r - 1). 



Since the theorem follows as above if jj+ 1 =(= 1, we seek to prove 

 that a value r can be found for which 



But a root of r^ s + 1 =l will satisfy 



**-*-!- 



only when 



131) l-r 2 -r = r. 



The desired value of r certainly exists if p" -f- 1 > 3. But if p* = 2 7 

 we have a = 1, whence the equation 131) has the single root r = 1 

 in the 6r.F[2 2 ]. The theorem has therefore been proven for all cases. 



150. Theorem. Excluding tlie case m = 3 ; p s = 2, $& 

 contains a substitution leaving one index fixed and not reducing to the 

 identity. 



By 149, I contains a substitution S =|= 1 which replaces ^ 

 by a function of the form % t + 2 Hence 



*1 a 



where S^ is a substitution of H w ,^, s of the form 



