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



principal in the grouped group, and the only principal 

 substitutions in the /- i derived groups 



Q(l, QiG, OjG, • . 

 will be QP QiP Q,P &c., where P is any principal of G. 

 This gives only 



(/-i)R, 



principal substitutions in the /- i derived groups. 



If the partition of N be not of the form N= A- A«i &c., 



no substitution Q, can be principal, and there will be only 



(/-i)R, 



principal substitutions in the /- i derived groups. 



33. Let X be the number of principal substitutions @ 



of g which have the form, (31), 



/e=:Afl!i + B6i + 0Ci+ • . + J/i; 



then there will be l-X- 1 non-principal substitutions in ff, 



besides unity. 



The number of principal substitutions in the grouped 



group will be 



'R^. + \k-j-{l-\- i)R;.= /A;+ (Z-\)R;t• 

 When \=o, whatever be the form of the partition of N, 



there will be only 



principal substitutions in the grouped group. 



A grouped group has its normal fo7in, when it begins 

 by the powers of a principal substitution. 



The number of different groups G of theorem C, formed 

 on the same partition of N, with which the group can 

 commence, is 



This is the number of repetitions of the same grouped 

 group that we shall make by selecting every one of the W 

 groups of theorem C for our group G, without making 

 any change in the auxiliary group g, or in the system of 

 exponents in the denominators of the /- 1 derivants Q, 



