GROUPS AND MANY-VALUED FUNCTIONS. 307 



Let us next construct on 0', in the same manner^ the 

 substitution 



a'= 



PkPk " PI 



Pi Pi ' • Pa Qa+l 



We shall have 



PlPT ' • pi 9a+l ■ ' ' 



for in order that 00' may be a substitution of g, it is 

 required that , 



kXjx •■ p ■-' 

 whence we must have 



PkPk --Pi " 



whereby we obtain the product Q,Q' above written. 



This proves that the substitutions 

 QQ'.(QQ')-- 

 which are formed on those of g compose with unity a 

 model group, for the reasons by which ^ is a group. 



Wherefore, whatever be the system of exponents which 

 we employ in the denominator of Q. Q,' &c., the same 

 system being used in them all, and whatever be the auxi- 

 hary group g, we have always a grouped group of kl sub- 

 stitutions, consisting of G and I- i derived derangements 

 of G. 



Let us endeavour to enumerate the groups thus con- 

 structed which shall have their principal substitutions of 



the form 



fp = Aa + Bb + Cc+ • • +Jj 



of the principal substitutions of G. 



This will restrict the number of auxiliary groups g that 



we can employ. 



^o. Let rt 



KfiTTa • • a 



be a circular factor of the order A in any one of the 



substitutions of g, and let us suppose it formed with h of 



the a elements of 



a + b + c+ • ' +J, 



