46 On the Theory of Groups. 



tion of three letters ; the group 



1, a, ^, y, ^, e 

 may represent a gi'oup 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 Caustics by refrac- 

 tion of a Circle." 



It is, I think, worth noticing, that if, instead of considering 

 a, yS, &c. as symbols of operation, we consider them as quan- 

 tities (or, to use a more abstract term, ' cogitables ') 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 



ic-{-aa. + b^+ ... 

 Thus, in the system just considered, 



{w + au + b^ + cy + d8-\- ee) {w' + a'u + b'fi + c'y + d'B + e'e) 

 =W-hAa + B/3 + C7 + DS + Ee, 

 where 



W = wiv' + ab' + a'b + cc/ + dd' + ee' 



A=wa' + w'a + bb' + dc' + ed' + ce' 

 'B—wb'+ w'b + aa' + ed-i- cd' + d^ 

 C = wc' + w'c + da' + eb' + bc^ + ae' 

 D = wd' + w'd+ea' + cb' + acl + b^ 

 ^=106' + w'e + ca' + db' + bd -\- ad'. 



It does not appear that there is in this system anything ana- 

 logous to the modulus w^ -^ x"^ -^ y'^ -{■ z^ , 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 alge- 

 braical equations, but also with that of 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. «, ^> y, «^, /3«, a7> y^y ^y, y^> «^y> ^7«^ 7«^> 



u^a, (dy/3, yay, «^y^, ^y/3u, 

 where 



u' = l, ^2=1, ^2 = 1, (/3y)3=l, (ya)3 = I, m' = h 



(«/3y)2=l, (^y«)2=l, (y«^)2«l; 



