ON THE THEOllY OF GROUPS, 
126 
[125 
If, on the other hand, a 2 =l, then it is easy by similar reasoning to show that we 
must have a/3 = /3a, so that the group in the case is 
1, a, a/3, (a 2 = 1, /3 2 = 1, a/3 = /3a) ; 
or if we represent the group by 
we have the table 
1> a, /3, y, 
1 a /3 y 
1 
a 
/3 
7 i 
a 
1 
7 
0 1 
/3 
7 
1 
a 
7 
/3 
a 
1 
or, if we please, the symbols are such that 
a 2 = /3 2 = y 2 = 1, 
a = /3y = y/3, 
/3=7« = a/3, 
7 =a/3 = /3a; 
[and we have thus a group essentially distinct from that of the system of roots of 
the ordinary equation ¿r 4 — 1 = 0]. 
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 
a (n) = 
c 2 
n 
/3(n) = - 
C 2 + n 
1 + n 
7 (n) = - 
C 2 ( 1 +n) 
c 2 + n ’ 
then a, /3, y form a group of the species in question. So in the theory of quadratic 
forms, if 
a (a, b, c) = (c, b, a) 
/3 (a, b, c) = (a, — b, c) 
y (a, b, c) = (c, - b, a); 
although, indeed, in this case (treating forms which are properly equivalent as identical) 
we have a = /3, and therefore y = 1, in which point of view the group is simply a 
group of two symbols 1, a, (a 2 = 1).