274 K.EV. T. p. K.IRKMAN ON THE THEORY OF 



XXI t. — On the Theory of Groups and many -valued 



Functions. 



By the Rev. Tho. P. Kirkman, M.A., F.R.S. 



Eead April i6th, 1861. 



§ 1. 



First principles : Factor's of a substitution : Permutable 

 substitutions. 



1. Let G denote the sum of k different arrangements 



Ai A2 • • A.j^ 

 taken from the 1 • 2 • 3 • • N permutations of the N elements 

 1,2,3- -N-i-N. 

 Every pair A,„ A^ of these k permutations gives a sub- 

 stitution, which is written 



A 



-"■TO 



and which has reference to a subject B, on which we 

 operate. This subject is any permutation of the N ele- 

 ments. The effect of the substitution on B is to exchange 

 in B, for any letter a, that which stands above a in the 

 substitution. We express that C is the result by writing 



f^B = C. 



A„ 



If the k arrangements are such that every result of 

 operation by any substitution made with any pair on any 

 arrangement is an arrangement of the system, they form 

 a group of permutations. 



