276 REV, T. P, KIRKMAN ON THE THEORY OF 



lohich you operate on A„, and at the same time write down 

 the consecutive elements of A^ in the same order in which 

 you see i 2 3 • -N disposed in A„. 

 For example^ in the group 



12345 

 31452 

 43521 

 54213 



25134, 



we have 



31452 -31452 = 43521 = (31452)^ 

 31452 -43521 = 54213 = (31452)' 

 31452. 54213 = 25134= (31452)* 

 31452. 25134= 12345 = (31452)^= (1) 

 31452.12345 = 31452 = 31452. 

 4. Every substitution A^ is obtained by operating on 

 unity with cychcal permutations of certain circles of ele- 

 ments, which are called the circular factors of the substi~ 

 tutiou. For example, the substitutions 



654213, 312564, 465132, 

 may be written 



^3452i, 321,564. . 41,62,53 

 163452^ '' 132,456^ '' 14,26,35^ i> 



of which the first has a circular factor of six elements; 

 the second has two factors each of three elements ; and the 

 third has three factors of two elements. 



It is usual to say that the first has a factor of the sixth 

 order, the second has two [factors of the third order, and 

 that the third has three factors of the second order. We 

 speak also of the order of the substitution 9, which is the 

 number of its dififereut powers, (1 d&^- • • •). 



It is well known that the order of a substitution is the 

 least common multiple of the order of its circular factors. 



It is useful to be able to see the circular factors without 

 losing sight of the form of the substitution in a model 



