324 
[690 
690. 
ON THE THEORY OF GROUPS. 
[From the Proceedings of the London Mathematical Society, t. ix. (1878), pp. 126—133. 
Read May 9, 1878.] 
I recapitulate the general theory so far as is necessary in order to render 
intelligible the quasi-geometrical representation of it which will be given. 
Let a, /3,.. be functional symbols each operating upon one and the same number 
of letters, and producing as its result the same number of functions of these letters. 
For instance, a (x, y, z) = {X, Y, Z), where the capitals denote each of them a given 
function of (x, y, z). 
Such symbols are susceptible of repetition and combination; 
a?(x, y, z)= a(X, Y, Z), 
£«(«, y, z)=/3(X, Y, Z), 
or 
in each case equal to three given functions of (x, y, z)\ and similarly for a 3 , a'“/3, etc. 
The symbols are not in general commutative, a/3 not =/3a; but they are associative, 
a/3.7 = a. /3y, each = a/3y, which has thus a determinate meaning. 
Unity as a functional symbol denotes that the letters are unaltered, 
1 0, V, z) = (x, y, z) ; 
whence 
la = al = a. 
The functional symbols may be substitutions; a{x, y, z) = {y, z, x), the same letters 
in a different order. Substitutions can be represented by the notation 
the 
substitution which changes xyz into yzx, or, as products of cyclical substitutions, 
a , = {xyz) (uw), the product of the cyclical substitutions x into y, y into z, 
z into x, and u into w, w into u, the letter v being unaltered.