364 REV. T. p. KIRKMAN ON THE THEORY OF 



143256 X2i3546( = )4i3526; 

 and we complete the group of eight thus : 

 123456 143256 «i 

 <f> 413526 213546 «2 

 <A' 543216 523416 «3 

 </>' 253146 453126 «4- 

 This^ as well as the group of sixteen preceding, is one of 

 the groups enumerated in theorem Gr, (26). 

 The partition of N is here 



6=4- 1-1-1 '2, and K=4. 

 77. Whenever the order K of the group i-H^-f^*- • is 

 even, the didymous factors agag[=^)<^^'^ are always two of the 

 same form, but those of <^2m+i ^^g f^Q qJ^ different forms. 

 Thus «i and a^ are of the form 



6 = 2 • 1 -I- 1 • 4= A« + Bb, 

 whilst fl!3 and a^ are of the form 



6 = 2-2-1-1 '2 = Aa + Bb, 

 When the order K of the group is odd, the didymous 

 radicals of the system are all of one form. 



An important point is here to be noticed concerning the 

 radicals of such a group as the one last written, formed on 



the partition 



N=A«-t-i.2, 



where the group i,(j),^'^-- has two letters undisturbed. 



The derived derangement EG of theorem G, (26)^ will 



consist of radicals having the same two letters undisturbed. 



But if we exchange those two letters, writing, for example, 



above, 



146253 a'l, for 143256 «! 



216543 a\ 213546 % 



526413 a's 523416 «3 



456123 a\ 453126 «4. 



we have still a system of didymous radicals sucb that the 

 product of any two is a power of 0. The form a'l a\ is 



6 = 2.2+ 1 »2, 



