640 
ON THE THEORY OF GROUPS. 
[887 
only one line to each of the points. We may have through two points a, b only 
the line ab of the given colour; this is then a double line regarded as drawn from 
a to b and from ft to a; and there is thus one and only one line of the colour 
from each of these points and to each of these points. The condition implies that 
the lines of a given colour form either a single polygon or a set of polygons, with 
a continuous currency round each polygon; for instance, there may be a pentagon 
abccle, meaning thereby the pentagon formed by the lines drawn from a to b, from 
b to c, from c to d, from d to e, and from e to a. An arrow on one of the sides 
is sufficient to indicate the currency. In the case of a double line we have a 
polygon of two points, or say a digon. 
There is a further condition which, after the necessary explanation of the meaning 
of the terms, may be concisely expressed as follows: Each route must be of 
independent effect, and (as will readily be seen) this implies that the lines of a 
given colour must form either a single polygon or else two or more polygons each of 
the same number of points: thus if a = kn 1 , they may form k argons; in particular, 
if « be even, they may form ^8 digons. 
To explain the foregoing statement, first as to the term “ route.” I denote the 
several colours by capital letters, R = red, G — green, B = blue, &c. Any capital or 
combination of capitals determines a route; R means go along a red line; RRBG, 
go along a red line, a red line, a blue line, a green line, and so in other cases. 
Given the starting point, or initial, the route determines the several points passed 
through, and the point arrived at, or terminal: thus aRRBG = abefk, — k, means that 
the route RRBG leads from a through b, e, f to k, viz. that the red line from a 
leads to b, the red line from b leads to e, the blue line from e leads to f and the 
green line from f leads to k. We may give in this way the Itinerary, or write 
simply aRRBG = k, meaning that the route leads from a to k. We may of course 
write R 2 for RR, and so in other cases. A single capital, as already mentioned, is 
a route, but it may for distinction be called a stage. A stage, and thence also a 
route, may be reversed; R- 1 means go along the red line drawn to the point; 
if aR = b, then bR~ l = a; and so if aRRBG = abefk, = k, then lcG^B^R^Rr 1 = kfeba, 
= a ; R-'R^ = R~ 2 , and so in other cases. 
The effect of a route depends in general on the initial point: thus, a route may 
lead from a point a to itself, or say it may be a circuit from a; and it may not 
be a circuit from another point b. And similarly, two different routes each leading 
from a point a, to one and the same point x, or say two routes equivalent for the 
initial point a, may not be equivalent for a different initial point b. Thus we 
cannot in general say simpliciter that a route is a circuit, or that two different 
routes are equivalent. But the figure may. be such as to render either of these 
locutions, and if either, then each of them, admissible. For it is easy to see that 
if every route which is a circuit from any one initial point is also a circuit from 
every other initial point, then two routes which are equivalent for any one initial 
point will be equivalent for every other initial point. And conversely, if in every 
case where two different routes are equivalent for any one initial point, they are 
equivalent for every other initial point, then every route which is a circuit from any