314 REV. T. P. KIRKMAN ON THE THEORY OF 



_ We see then that the derivates 



will be reproduced by a different system of exponents^ in 



the form 



QiG Q'lG Q'\G" 



where Qi is the (i+i)''' arrangement of Q, i being a 



multiple <k of all the quantities 



AA^ BB J^£ 



Ai A2 Bj B2 Ji J2 

 where 



AA-. B1B2.. J1J2 



are the orders of all the circular factors in the group g. 



Let M be the least common multiple of all these frac- 

 tions. 



k . . 



• There are in QG ^nr substitutions Qi, each of which 



M 



can be constructed as a derivant and will give a derivate 

 (35) QjG identical with QG ; that is, we shall, by using 

 every possible system of exponents, repeat the same 

 grouped group 



times. 



It is therefore necessary to divide (S), the sum of our 

 constructions, by this number 



A 



M- 



It is evident that the partition of a + b-\-c+ ' • +j, 

 which gives the group (/, is to be treated as the separate 

 partitions 



a = Aiai + A.2a.2+ • ■ 



Z» = Bi6i + BA+--. 



if we wish to enumerate the equivalent groups ff. 



For the rest, g may be any group whatever, which has 



Ai A2 • • all divisors of A, B^ B2 • • all divisors of B, &c. 

 37. We have demonstrated the theorem following — 

 Theorem H. Let any partition of N be 



