370 REV. T. P, KIRKMAN ON THE THEORY OP 



radicals of a certain substitution ^. We have (Art. 76), 

 ~ = ax)b = <j) = bc = cd=:ide, &c. 



hajj^c, cbc=d, dcd=e, &c. 



Hence all tlie series aj)c- • can be proved to be present 

 inH'. 



Let yS be any primitive root of N - 1 . Then one of the 

 sets of N - 2 radicals is always obtained by writing as the 

 first line the substitution 



made with N-2 elements^ and completing under each 

 element the vertical circle 



j^iv-3^iv-4. .^2^^ (mod. (N-i)). 



We then add the elements N and N - 1 to each of the 

 radicals so obtained. 



If under any one {b) of the - (N - 2) thus found, which 



have no letter undisturbed, we write the didymous radicals 



of 



^=2345..(N-i)iN, 



viz. 



iN-i N-2- • 432N («o) 



N-i]Sr-2N-3.- 321N {bo) 



&c., we always find more than one of the quotients 



^ ^ ^ t 



tto Oq Co «o 



which are substitutions of the order N. 



b 

 Let (f) = bao = ~ be of the order N ; then we have 



present in H' the series 



a^cd' ' 

 of N didymous radicals of ^ ; and as no two of bed' - end 

 in N, and as no two end with the same element, they are 

 found one in each of the N - 1 groups li!^ A'j A'g • • j and the 

 N - 1 derivates of G' above formed are 

 6G', cG', ^G', &c. 



