388 KEY. T. P. KIRKMAN ON THE THEORY OF 



c=3 and c = ^. And it is easy to satisfy one's self that 

 the equation Q. is satisfied when ISr= ii, by taking c=y or 

 c = S for the determination of A in ^^; that is, Q, is satis- 

 fied when N = 7 by 



B = - 1 and C = - 2, 

 or by 



B= -2 and C= - 3, 



and when N = 1 1 it is satisfied by 



B=:5 and C = 4, 

 or by 



B = 3 and C = 2. 



94, But there is no need to attempt the eliminations to 



which the condition Q, invites us. We can much more 



readily settle the matter by tactical considerations; that 



is, we can determine whether a given value of c in 



(jii =r i-i -f A (i-i - i«^^-^0 = Bi-1 - Ci^^"^ 

 gives one of a system of didymous radicals which will 

 complete the group in question. 



If the equation Q, be satisfied by the system, it is satis- 

 fied when ^™=:/3'*=/3''= 1. We must have, putting k=o, 

 and/3™=i=/3% 



4>i{l + h)(j)i{i-k){ = ){i + z)4>l3''{i-y), 

 whence comes 

 ^i{i + k) cj,i{ = ){i + z) cf>mi-y)){i + h){ = ){i + z) cl>mi-t)). 



N- 1 

 As there are N values of h and only of a?, there 



will be one or more values of h that will introduce /3*'= 

 ^•^=1 into the right member. If then the group exists, 



^i = (f)i{i + h)(})i{ = ){i + z)(}>{i-t){ = ) (j>{i-t)+z=\''. 



This affirms that -v/^i, the A"' power of the substitution 

 of the W^ order, A,, which is determined by the circular 

 factor (pi, difiers from a certain cyclical permutation (f>{i—t) 

 of (f)i, only by the addition of the constant z to every ele- 

 ment ; that is, if 



^i = abcde- -, 

 '^jri — a'b'c'd'e' ' <•, 



