GROUPS AND MANY-VALUED FUNCTIONS. 309 



in O is a divisor of A, and that every circular factor in 

 © made with the a or 6 or c- • elements is of an order 

 which divides A or B or C, &c. © is here any substitu- 

 tion of g. 



And QP, whichever of the principal substitutions of G 

 P may be, is a substitution of the order of P, and a prin- 

 cipal substitution of the grouped group 

 G + QG + Q'G+.. 



31. There is nothing to prevent Q- from being also a 

 principal substitution of the group constructed. If it be, 



we have 



fQ = Aa + Bb + Cc+ ■ ■ +Jj. 



But the orders of the circular factors of Q, are those of the 

 factors of 0, on which Q, is formed. Hence © must have 

 the form 



z=a + b+c+--+j, 

 whence it appears that the partition of N on which G is 

 formed is 



^ = A'Aa + B-Bb + C-Cc+ ■ ■ +J-JJ. 

 When Q has the above form {/Q), every substitution of 

 the derived group Q,G is a principal substitution of the 

 grouped group ; and every principal substitution 6)„ of g 

 will give a derivant Q,^^ such that Q,,G has k principal sub- 

 stitutions. 



32. Let Qa be not a principal substitution of the grouped 

 group. The partition of N may or may not be of the form 

 N = A'Aa &c. above written. If it is of this form, Qa 

 which is not principal is constructed on ©a which is not 

 principal in g; and QaG will have only B,^. principal sub- 

 stitutionSj one for every principal of G. 



If the partition of N be N = A -Aa kc, and if the prin- 

 cipal substitution of g has circular factors of an order 

 below A made with the Aa^ letters, or of an order below 

 M made with the Mmi letters, no substitution Q will be 



