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



The other groups of the order 2(N - 2) are 



\M.%\y\ XaM'VXi"', \^M.%K\ &C.; 

 whence we see that if a be any one of the (N- i)(N-2) 

 didymous radicals added to K, 



is another, and we have 



whichever X.^ may be of the substitutions X^-^^ • • • 



Let a,J)j;,^' • be the didymous radicals of the group 

 M"„. If we write all the derivates 



b'Jj = b^ + b,,\ + bj^i + • ' 



C„ Jj = C^ + C^ A-j^ + C^ A2 + • • 



we have thus formed 

 since 



This derivate a„K will have for its final vertical row 

 the letter n which is undisturbed in M"^. We can 

 thus complete N - 1 derivates of K, which along with 

 K make 



(N-i)(N-2) + (N-i)(N-i)(N-2)=:N(N-i)(N-2) 

 substitutions, to every one of which (a) corresponds an- 

 other {13) such that 



88. It can be proved by a simple tactical argument 

 similar to that mentioned in Art. 81, that if a and ^ be 

 any two of the square roots of unity of the system of 

 groups of the order 2(N- 2), 



M"iM"a.., 

 (which radicals, including /*i*'^~^', /"■2*^^~^^ &c., when N - 2 

 is even, that is, when p>2, which we suppose to be the 

 case, make up the number 



