284 REV. T. p. KIllKMAN ON THE THEORY OF 



The elements of the second factor of the order A casi 

 be chosen among N — A in 



n(N-A) 



JTA . n(N- 2A) 



different ways. If a = 2, we have 

 1 HN 



1-2 (7rA)2 7r(N-2A) 

 different ways of choosing the 2A elements, where we 

 divide by 2, because either of the two circular factors may 

 be considered as the first chosen. 

 In like manner there are 



-1 ilN 



na ' {it Ay IT (N- Aft) 

 ways to select the a circular factors of the order A. 

 In the same way there are 



_1 n(N-Aa) 



nb{nBY (N-Aa-m) 



ways to select the b factors of the order B from N - Aa 

 elements, and 



1 n(N-Aa-B6) 



nc {ncy ji(n - Aa - b6 - Cc) 



ways to select from the remainder the c circular factors of 

 the order C. 



Consequently there are 



^ "^ =U 



nanbUc- -nj {nA)"inB)\ncY.-{njy 



different ways to choose the circular factors. 



We can write each of the circles of A elements in a 

 vertical row headed by its least element in n(A - 1) ways ; 

 whence there are 



{n{A-i)y{n(B-i)y{n{c-i)r--in{j-i)y=Y 



ways of writing the circles in vertical rows under unity 

 (Art, 4). Wherefore we have thus 



UV = 



nanb-.Uj A'^B^C'^.-J' 



