as depending on the Symbolic Equation 6" = 1. 



43 



Or, if we please, the symbols are such that 



Systems of this form are of frequent occurrence in analysis, 

 and it is only on account of their extreme simplicity that they 

 have not been expressly remarked. For instance, in the theory 

 of elliptic functions, if n be the parameter, and 



c c -\~n 



a{ii) = — fi{n) = — 





then a, /9, 7 form a group of the species in question. So in the 

 theory of quadratic forms, if 



a[a, b, c)=:{c, b, a) 



0{a, b, c)={a, —b, c) 



7(a, b, c) = {c, -b,a); 



although, indeed, in this case (treating forms which are pro- 

 perly equivalent as identical) we have a=/3, and therefore 7 = 1, 

 in which point of view the group is simply a group of two symbols 



Again, in the theory of matrices, if / denote the operation of 

 inversion, and tr that of transposition, (I do not stop to explain 

 the terms as the example may be passed over), we may write 



«=/, /3=tr, ry=I .tr=trl. 

 I proceed to the case of a group of six sjonbols, 



1, a, /3, 7, B, €, 

 which may be considered as representing a system of roots of the 

 symbolic equation 



^ = 1. 



It is in the first place to be shown that there is at least one 



