324 KEV. T. p. KIRKMAN ON THE THEORY OF 



whicli has the same number of horizontal and vertical 

 rows with Gr ; thus adding ©^G ©gG • • • 



The r- X derived groups will complete with G a group 

 of the order kr, and they are all derived derangements of G. 



For let 



(rK+cO)(yi^m+ci))(2/i7«.+ci))- • =<i> 



he any two of the kr substitutions, c^ c' being each <k. 

 The product ^^' is 



^^' = {if^f{a„, + c') + y'c^) {r^^{^„, + C) + y%) • • , 

 which is a substitution of the derived group 



(2/«+%„, + c)) (?/^+^(/3„, + c)) (2/^+%^ + c))'-, 

 viz. the substitution which has the value of c, 

 c=c' + y~^Ci = c' + y''~^Ci (mod. k) . 



42. We obtain thus the group, (Def. Art. 2), 



Jz=G + eiG + e2G+ . • +0,_iG. 

 If we take any other of the W groups of theorem C, 

 G' = aGQ-^ 

 we can form the group {©'^=0,0^0,"^) 



J'= G' + ©'iG' + ©'oG' + . . + ©"-^G' 

 equivalent to J, and we have thus W equivalent groups of 

 kr substitutions by this root y. 



Hence we have the theorem following — 

 Theorem I. If there be \ different roots y, of which no 

 one is comprised among the powers of another^, which fulfil 

 the condition of Art. (41), we can construct 



XW 

 {theorem C, 12) equivalent groups of the order kr, of the form 

 J = G + ©iG + ©2G+.-+©,_iG, (Art. 41). 



43. The number of groups constructible is much greater, 



if 



«/=E-i, >i, 



E being any one of the numbers ABC • • J. For we can 



add to the group 



^=(«m + c)(^»,+ c)- .(6,^ + c)- . 



