GROUPS AND MANY- VALUED FUNCTIONS. 311 



{29). Wherefore (S)^ the sum of our results, must be 

 divided by this number. 



34. We have to consider the effect of a variation in this 

 system of exponents. 



We can always take uuity for the first exponent in the 

 denominator of Q, of every circular factor in Q, ; for 



p^pf^pr-'pf~ p. p^-^^'p^'^^' ■' pf-""^'- 



If the a factors PiP-iP^- • make in Q. a^ circles of the 

 order A^, a.^ of the order Aj &c., and if the b factors make 

 ^1 of the order Bj and 62 of the order B2 &c.j the number 

 of different systems of exponents is 



A ff— di— B2— •• "DS— 61— 62— .. Qc— ci— C2— •'• Jj—ji—h— •• . 



for there are a-ai-a.^- • - of the a factors PiPz' • which 

 are not first in their circles, to which we can give at plea- 

 sure any one of the exponents 



1, 2, 3, • -A; &C. 

 The first derivant Q, is always constructed on &, a prin- 

 cipal substitution of g, and the denominator in Q is that 

 of all the derivants. This denominator is determined by 

 the form of 0, (28), 



f© = A16J1 + A2«2 + Asflfg + . • { = a) 



+ BA + ^2h + ^sh+ • • [ = b) 

 &c. 



35. The substitution Q' which gives the derived group 

 a'Gis 



Q>_ PIPI ■■ PlP^lPl '• Pi-- 



where 



' P^VIPI - - P'mPlPl - • P\ 



abc- -1 ^ Iq- -n 

 and 



lab- -m nl- -t 



are two of the circular factors of 0' of the orders r and s, 

 on which Q' is constructed, these factors being formed out 

 of the a elements in ©'. 

 The substitution [i < k) 



