TTieory of Groups depending on Symbolic Equation ^" = 1 . 35 

 where the laws of combination are 



^2=1, «2^/92 = ry2 = ^, 



^7=«, 7«=/S, «/3=r, 



fy/3=«5 = d«, a7=/3S = S/3, /S« = 7S = .a7. 



Observe that ■& is a symbol of operation such that ^^ = 1, and 

 that ^ is convertible with each of the other symbols a, /8, y. It 

 will be not so much a restrictive assumption in regard to S, as 

 a definition of — 1 considered as a symbol of operation if we 

 write d= — 1 ; the group thus becomes 



1. «> ^, y, — 1> — «> — ^^ —y> 



where 



«2 = ^2 = 72=_1, 



a =^7 = — 7/3, /8 = 7« = uj, 7 = «/3 = ^a. . 



Hence «, /S, 7 combine according to the laws of the quaternion 

 symbols i, j, k ; and it is only the point of view from which the 

 question is here considered which obliges us to consider the 

 symbols as belonging to a group of 8, instead of (as in the theory 

 of quaternions) a group of 4. 



Suppose in general that the symbols », /3 are such that 



then we find 



«'»=1, /3" = 1, u/3=/3u', 



'=B» 



and therefore if v = n, a" = «"*" or gj«(«"-') = 1^ whence 



m(s" — l)=0(mod.m) ; or since wis arbitrary, s"— 1=0 (mod. m), 

 an equation which, if m, n are given, determines the admissible 

 values of s ; thus, for example, if ?i:=2, and m is a prime number, 



then s = l ors = m — 1. The equation iit'^ff' = l3"»"' shows that 

 any combination whatever of the symbols «, /3 can be expressed 

 in the form /S'a'' (or, if we please, in the form aP^''). It is proper 

 to show that the assumed law is consistent with the associative 

 law, viz. that the expression 



can be transformed in one way only into the form ^aP. We in 

 fact have 



/3*a" . ^^' = 0' . «"/3'' . cc' = 0' . yS"*'"' . «'^ = /3*+'^«"'''+"; 

 and multiplying this by the remaining factor /S^a*, we have 



D 2 



