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



The following investigation of M. Matliieu's theorem s, 

 when N - 1 is the power of any prime number, will be 

 probably found simpler than either. 



We begin first with N - i^ a prime number. 

 By a theorem of Cauchy's, we can form with the N- i 

 elements 123' -(N-i) a group G of (N-i)(N-2) sub- 

 stitutions all of the form 



G = S{pi + c), 

 where p has any of the N - 2 values 



123- -N- 2, 

 and c any of the N - 1 values 



0123- •N-2, 

 by writing, for any element i of unity, 

 pi + c, (mod. N-i), 

 In this group G there are N - 2 substitutions ending in 

 (N- 1), which form a group g of the order N-2; and G 

 contains also g,., a group of the same order, in which the 

 element r is undisturbed. The substitutions of this group 



g^ are 



S {pi + r -pi') . 



Thus G can be written as the product of the group 



go=S{i + c) 

 of N - 1 powers, 



1+6 + 6''+ ••+6''-% 

 where 



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



and of N - 1 groups g^ of the order N-2, in each of 

 which one element is undisturbed. That is, 



The N - 1 didymous radicals of g^ are 



where 



a(pQ = 6, aQCQ = 6', aQdQ = 6^, &c, 



b(fCQ=6, bQdQ=6^, bf^eQ=6^, &c. 



CQdo=6, ^0^0=^^ &c. 



And these N - 1 radicals are the central powers of the 



