313 KEY. T. P. KIRKMAN ON THE THEORY OF 





Px vlvl •' A Pi p\ •• ft •• 



is the substitution Q'P', P' being the i"' substifcutiou in 

 order after unity in G. 



The derivate Q'G is also Q"G, (7). If we have formed 

 Q"G among our constructions made by varying our system 

 of exponents^ as the derivate Ql^, it will be possible to 

 write Q" in a form Q'l that differs from Q' only in the 

 exponents of the denominator. 



Now 



P, Vt' Pl-^'' ' ■ Pt:''"''' Pn Pt' ■ • pr''-'''- ' 



r and s being the orders of the factors of & . And this 



has not the form of Q,' unless 



ri = hK 

 si = kA 



or 



._hA_kA_ 

 r s 



In the same way we must here have 



._hjC_k^_ 

 ^// *// 

 where 7\Si • • are the orders of the circular factors of 0' 

 formed with the b elements r^^s^^- • , those of the factors 

 formed with the c elements^ &c. 



We conclude that i is not less than the least common 

 multiple M of 



AA _B]B _ CC _ 



and that it may be any multiple < ^ of this number M. 

 36, The group g is formed on the partition 



