304 KEY. T. P, KIRKMAN ON THE THEORY OF 



Z Pi p\ 

 563412 



321645 



This group Hg^G' + R'G' is equivalent with the pre- 

 ceding. We have, for example, 



H3=i25346Hi(i 25346)-!, 



\PIP.J ^^Plpj- 



In like manner, it may be proved that all the groups 

 are equivalent which we enumerate in the theorem fol- 

 lowing. 



Theorem G. Let 



N = Aa + B^H hJi 



A>2, A>B, B>C-., 

 and let k be the least common multiple of ABC • • J. TVe 

 can construct in this partition of N, 



HN 



Uj^kllallb- 'IIj 



equivalent groups of 2K substitutions, each of the form 



G + RG, where G is one of the groups of k powers {of 



theorem C), and RG is composed of k square roots of unity. 



§ 7- 



Grouped groups of the first class. 



27. Let 



N = AaB6+ • • +J;' 



A>B>C-»>J 



be any partition of N. 



A substitution P is a principal substitution of a group 

 formed on this partition, when it has the form 



fV = Aa + -Bb+-'+Jj, 

 which means that P has a circular factors of the A'^, b of 

 the B"' order, &c. ; and if the form of every other substi- 

 tution Q, of the group is 



/Q=A'a' + B'6' + C'c' + , 



