Let us suppose 



308 KEV. T. p. KIRKMAN ON THE THEORY OF 



The substitution Q, which we form on O will have the 

 circular factor 



pIp'I-'pV 



Pi = a^b^c^ • • 



..Pe — agbeCf ■ 

 where the number of the elements a,.b^c^ • • is A. 



The substitution P which is written under unity in the 

 group G may be represented by 



Jm 



for P puts for any element i,„ in a circular factor the ele- 

 ment jjn which follows in that factor, ^,„ or g„„ &c. 

 The substitution QF is of this form : 



OP- b^C^^cr • ■ 



a>cbf,c„d^ 



It is plain that this circle will be closed when the circle 

 abed- • of the A'* order, and the circle KfjbTra- -6 of the 

 order h, are both completed ; and that the order of this 

 circle of Q,P will be the least common multiple of A and 

 h. There will be A of these circles in UP beginning with 

 bK, c^, d^, &c. 



It follows from this view of the circular factors of Q,P 

 that the order of this substitution is the least common mul- 

 tiple of the orders of S and P. 



Let us suppose that P is a principal substitution of the 

 constructed grouped group, which implies that Q,P shall 

 have no circular factor of order above A made with the 

 Ka elements, nor of order above B made with the B5 

 elements, &c. 



It follows that h the order of the circular factor kixtto- ■ ■ 6 



