AS DEPENDING ON THE SYMBOLIC EQUATION 6 n = 1. 
129 
125] 
or, what is the same thing, the system of equations is 
1 = fia = afi = y I 2 = S 2 = e 2 , 
a = /3 2 = 8y = e8 = 76, 
/3 = a 2 = 67 = 7S = Se, 
7 = ha = e/3 = /3S = ae, 
S = ea = 7/3 = ay = fie, 
e = 7a = S/3 = /S7 = aS. 
An instance of a group of this kind is given by the permutation of three letters; 
the group 
1, a, fi, 7, 8, e 
may represent a group of substitutions as follows:— 
abc, cab, bca, acb, cba, bac 
abc abc abc abc abc abc. 
Another singular instance is given by the optical theorem proved in my paper 
“ On a property of the Caustic by refraction of a Circle, [124].” 
It is, I think, worth noticing, that if, instead of considering a, fi, &c. as symbols 
of operation, we consider them as quantities (or, to use a more abstract term, ‘cogi- 
tables’) such as the quaternion imaginaries; the equations expressing the existence 
of the group are, in fact, the equations defining the meaning of the product of two 
complex quantities of the form 
w + aa + bfi + ... ; 
thus, in the system just considered, 
(w -P aa -p bfi -P cy -P d8 -P ce) (w -P a a -P b fi •Pc73~dS-p6e) = W + A a -p JBfi 4- Gy -P D8 + Ee, 
where 
W = ww' + ab' + a'b + cc' + dd! + ce', 
A = wa! + w'a+ bb' + dc' + ed' + ce', 
B = wb' + w'b -P aa' + ec' + cd' + de', 
C = wc' + w'c -P da' + eb' + bd' + ae', 
D = wd' + w'd + ea' + cb' + ac' + be', 
E = we' + w’e + ca' 4- db' + be' + ad'. 
It does not appear that there is in this system anything analogous to the 
modulus w 2 + x 2 + y 2 + ¿ 2 , so important in the theory of quaternions. 
I hope shortly to resume the subject of the present paper, which is closely 
connected, not only with the theory of algebraical equations, but also with that of 
c. 11. 17 
Safe