330 REV. T. p. KIRKMAN ON THE THEORY OP 



Wherefore we have this 



Theorem K. Let J be any group of the order kr formed 

 on the partition of N elements 



N=Aa + B5+ . • +Cc, 

 comprising any a equivalent groups made each of A ele- 

 ments, any b equivalent groups each of B elements, the 

 equivalent groups being either iv'ith or ivithout repetition of 

 their substitutions of the order kr ; and let T be any group 

 whatever of the order I made with 



a+b+c+ ' • +j 

 elements, of which the a first vertical rows contain only the 

 a elements; the h next vertical roivs contain 07ily the h 

 elements, ^c. 



Every pair of groups JF gives a grouped group of krl 

 substitutions, ivhose elementary groups are the groups of 

 the order kr ivhich compose J. 



The equivalent groups of the order kr may be any of 

 the groups formed on the partition of N which have been 

 enumerated in the preceding theorems; that is, any one 

 of them may be determined either by one or by many cir- 

 cular factors. 



48. It is difficult; to determine how many of the grouped 

 groups of krl substitutions thus formed may be presented 

 as grouped groups of At^ substitutions formed by the pro- 

 cess of Artt. (37, 38). 



But there is an enormous number of them which cannot 

 be so presented. If, for example, we form by theorem Gr 

 (26) a group G + RGr of the sixth order on the partition 



9 = 3-3. 

 and then add to it either two or five derived derangements 

 by the auxiliary groups 



P=i23, or P'=i23 132 

 231 231 213 



312 312 321, 



