130 
ON THE THEORY OF GROUPS, &C. 
[125 
the composition of quadratic forms, and the ‘irregularity’ in certain cases of the 
determinants of these forms. But I conclude for the present with the following two 
examples of groups of higher orders. The first of these is a group of eighteen, viz. 
1, a, /3, y, a/3, /3a, ay, ya, /3y, y/3, a/3y, /3ya, ya/3, a£a, /3y/3, yay, a/3y/3, /3y/3a, 
where 
a 2 = 1, /3 2 = 1, y 2 = 1, (/3y) 3 =l, (ya) 3 = 1, (a/3) 3 = l, (a/3y) 2 = l, (/3ya) 2 = l, (ya/3) 2 = l; 
and the other a group of twenty-seven, viz. 
1, a 2 , 7, 7 2 > 7«> a 7> 7« 2 ; a2 7> 7 2a > a 7 2 > 7 2 « 3 , 
aya, ay 2 a, a 2 ya, a 2 y 2 a, ay a 2 , ay 2 a 2 , a 2 ya 2 , a 2 y 2 a 2 , yay 2 , ya 2 y 2 , y 2 ay, y 2 a 2 y, y 2 aya 2 , yay 2 a s , 
where 
a 3 = 1, y 3 = 1, (ya) 3 = 1, (y 2 a) 3 = 1, (ya 2 ) 3 = 1, (y 2 a 2 ) 3 = 1. 
It is hardly necessary to remark, that each of these groups is in reality perfectly 
symmetric, the omitted terms being, in virtue of the equations defining the nature 
of the symbols, identical with some of the terms of the group: thus, in the group 
of 18, the equations a 2 = 1, /3 2 = 1, y 2 = l, (a/3y) 2 = 1 give a/3y = y/3a, and similarly for 
all the other omitted terms. It is easy to see that in the group of 18 the index 
of each term is 2 or else 3, while in the group of 27 the index of each term is 3. 
2 Stone Buildings, Nov. 2, 1853.