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



w= , 



Ui^nanbHc'-nj a^b^c^.-p 



where ^j^ is the number of integers, unity included, which 

 are less than k and prime to it. 



13. Any two G G' of the W groups are equivalent, that 



is, we have 



G'=0G0-i. 



Let 



m^m=^m^"m„, 



be the m circular factors of G of the order M, and let 



m\m\m\ • • m'^ 

 be the m circular factors of G' of the same order. We 

 have 



^ _ a\ a\ • • a'^ b\ h\ • • h\ c^. c'.^ -'C',-- _ 



a\0-%- ' a a ^1 ^2 • • ^6 CyC<^' 'Cc' ' 



For example, let 



N = 8 = 6.i+2.i=:Aa + B6. 



Two of the W groups are 



12345678 12345678 



36487215 86714532 



42851637 25381476 



86573241 64728135 



52714683 51362874 



76138254 48756231 



^ a\h\ 18265437 o /: 

 ^ = ^=13485726 = ^3825746. 



G'=0Ge-i= 13825746 G14275863. 



§4- 

 Modular groups formed with N elements of an order superior 

 to N, which contain the powers of a substitution of the 

 W^ order. 



14. We have proved, theorem B(i2), that there are 



— ^ equivalent groups of the powers of a substitu- 



tion of the N''' order made with N elements. 



