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



<^' = B/3«(i - k)-^ - C/3" (i - kf^''-'' + k, 



where B and C are given functions of the number c chosen. 



And it is evident that if ^, be any one of the entire system 



N-i 

 of N radicals, 



is another, and that 



6" <^,=(pj\ 



N-i N- 1 

 Add now, to the N h N substitutions in G 



' 2 2 



and the added didymous radicals, N - i cyclical permuta- 

 tions 



of every radical 0. It is plain that we have thereby added 



N- 1 

 (N - 1 ) N substitutions 



ej>, e^cp,^ e^cj^,,,, &c. 



All that is further required, in order that the N + 1 



N- 1 . . 



N- substitutions thus formed should be a group, is 



N-i 

 that the product cf)^(f)^^ of any two of the N didymous 



radicals should be either another or a cyclical permutation 

 of another -, that is, we must have 



0^<^^^ = (j)"e" = OY = (f>" + b. 



There is no difficulty in ascertaining this point algebrai- 

 cally, except the usual one of elimination. 

 The condition to be satisfied is 



{B{^^ {i - k)y'-' - c(/3'' {i - k)y-^'--'^ + k} 



{ = ){B(^^{i-y)y-^-Ci/3ni-y)y^-'^ + z} (Q), 

 and it is required, in order that the group exist, that <», y, z 

 be determined in terms of BCMmw, and that c, of which 

 B and C are given functions, should be found by an equa- 

 tion independent of mnkh. 



93. The only important point is to ascertain the result- 



