360 llEV. T. p. KIRKMAN ON THE THEORY OF 



The theorem F (page 293) requires no correction. 



74. It is not necessary for me to determine the form of 

 N when p > 1 . But I suspect that the only possible form is 



where n is a prime number. 



We know that, in all the groups of theorems D and E, 



no substitution 



pi + c 



is of the W^ order, even when p > 1, if c be a factor of N ; 

 for it is found in PG = Pi + P2 + P3+ • •, a derived derange- 

 ment of 



G=s{i+c) = i + e+e''+-., 



and has the same vertical rows with G, of which every 

 substitution can be written, (6, page 279), 



9 ~Vy; 

 wherefore if 6 and P^ are of both the N** order, 6"^ and P^ 

 must be both of one order. When ]) is one of the p num- 

 bers, P^=j9i+ 1 is of the W^ order, like 6=i-\- 1; and as 

 O"', when a is not prime to N, is of an order below the N**, 

 P„ is not of the W^' order. Therefore 



can be raised to the power unity, 



P;/( = ){pi + aYi{=\fH Jr^^^^a{=)i, 

 p 1 



where r/<N, and where p^^t, (mod. N). 

 And as 



P*"! — p 



- — ->Q for rj<'N, 

 p- 1 



we must have 



p'^^ -p N 



p- 1 a' 

 which may be any factor whatever of N, < N, > 1 . 



I think it will be proved that N and a must of necessity 

 be powers of the same prime number. 



75. It is worth remarking, by way of supplement to the 



