328 REV. T. p. KIRKMAN ON THE THEORY OF 



of -T- substitutions of A elements, of which every substi- 

 tution is repeated -r- times ; of b equivalent groups each 

 of ^ substitutions which are repeated :^ times, &c. 



If the a equivalent groups were all made with the same 

 A elements 123- .A, they could be written 



^-f-Xt)' 



PoPiPi' ' being the circular factors of the A^'' order deter- 

 mining the a groups g^yigi' • 



The a groups are in fact formed with the different sets 

 of A elements 



1 23 .. A 



ii2i3i' • Aj 



1 22332 ■ ■ -t^2 



and we can evidently still write 



„ Pm ( Pm \ 



or 



„ Pm Pm 



^'''P,~PJ'' 



if we understand that the derived group . - 



pj" 

 has lost both the elements p^ and their order ; but that 

 the derangement 



Pm 



has preserved the elements p„^ and has exchanged their 

 order for that of p^. 



With this understanding the derivate 



PlUr, 



Po^' 



