368 REV. T. p. KIKKMAN ON THE THEORY OF 



substitutions of G along with the (N - i ) (N - 2) didy- 

 mous radicals of h^^ • • made with N - 1 elements, 1 23 • • 

 (N-i). 



If we add N as a final element to each of these 2(N- 1) 

 (N-2) substitutions, G so augmented is still G' a group 

 of (N- i)(N-2), and each of the augmented groups h\. is 

 still a group of 2 (N-2) substitutions, containing the aug- 

 mented g'^. 



And it is easily proved that every pair £py of the first 

 N - 1 elements is found once, and once only, in the order 

 xy in the c"' and d"'' vertical rows of H written in a 

 column, and once, and once only, in the order yw in the 

 same two vertical rows of H, whatever c and d may be 

 <N. 



If further we exchange in all the didymous radicals of 

 g'r the elements N and r, they remain still didymous radi- 

 cals of the augmented group ff\.. 



Let the system H so modified become H'. 



It remains true of the group li'^'^i\ • • , so modified, that 

 if h\ be any one of them, the others are 



eh:,e-^, evi',e-\ e^N.e-^ &c., 



where 



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



that is, all the didymous radicals in the modified system 

 H' are such that if a be any one of them, 



is another, similar to a, whatever c may be. 



But the (N- i)(N-2) didymous radicals of the system 

 H' are now alternately of the two forms, 



N-2 



N = 2 1-1-2 =:Aa + B6, 



2 



N 

 N = 2 — , =k.a; 

 2 ' 



that is, they are of the forms always read in the didymous 



radicals of the powers of a substitution of the order N. 



