GROUPS AND MANY-VALUED FUNCTIONS. 313 



a + b+ • ■ +j=Ayai + A^a^ + • • ( = «) 

 + 



Ai > Aa ■ • Bi > B2 • • Ji > Jo • • ; 

 and its principal substitutions have a^ factors of the order 

 Ai, bi of the order Bi, &c. 



The group g may be of the form 



y + 0y + 6f/+ • -, 

 where 7 is a group of powers of a principal substitution 

 of ^. 



If there be in g any substitutions containing circular 

 factors of orders different from the orders Ai Ag* -BiBj- • 

 seen in the principal substitutions, those orders will all 

 be divisors of A, of B, &c,, by our hypothesis (30). 



Our object is to determine the number of repetitions of 

 grouped groups due to changes in the system of exponents 

 of Q, the same G- and g being retained. 

 If the system of derivates 



QG, Q'G, Q"G, •• 

 made by one set of exponents, be repeated as 



Qi G, Q, 1 G, Q,i" Gr, •- 

 made with another set^ 



Qj will be the (1 +iy'^ arrangement of QG, 

 Q'l „ {i+iy „ Q'G, 



Q"i „ {i+iy „ Q"0; 



and although QG and Q'G may be repeated, Q"G will not 

 be repeated, unless there be in Q"G a (i+i)'^ arrange- 

 ment, in which « < A" is the least common multiple of 

 AA BB CC 

 r s r^ s^ r^^ 5„ 



rs • • r^s^ • • r,^s,, • • being the order of the circular factors of 

 0" on which both Q" and Q"i are constructed, ©" being 

 any substitution of g. 



SER. III. VOL. I. ss 



