502 Proceedings of the Royal Society 
did not at the time pursue the subject, as I found that it was more 
complex than it appeared at first. 
Mr Kempe's paper in Nature (February 26, 1880) has recalled my 
attention to the subject, and some simple modes of treating the 
question have occurred to me. The germs of them are in what I 
have said above, and they show one easily how to proceed to colour 
any map. A sketch only of one of them is now given. 
Begin by making as above stated a companion diagram, putting 
points for districts, and lines joining them for common boundaries. 
Then by introducing (in any way) as many new joining lines as possible 
(but so that no two intersect) the diagram is divided into three-sided 
compartments. 
Next, make all of these compartments four-sided by taking a number 
of new points, each on a joining line. The whole set of points can 
now be lettered A and B alternately, because two colours suffice for 
a map whose boundaries meet in fours. But let the intruded points 
be lettered a and b, instead of A and B respectively. 
Now perform the same operation in a second way, differing every- 
where from the first, and call the newly intruded points a and J3 
instead of A and B. 
Buies are laid down for carrying out these operations ; but they 
require too many illustrative cuts to be given here. 
Then any one triangular compartment will appear in two essentially 
different forms : for instance, with its intruded points it may read 
(in the two cases, taking the corners in the same order) B, a, B, A and 
B, A, /?, A. Now superpose the two figures, lettering included, and 
attend to the order of the two letters at the same point. We have, 
from the instance above, the compound reading (attending now to 
the corners only) BB, BA, AA, of which the separate terms are 
necessarily different. Hence every point in the figure is lettered 
differently from all that are joined to it, and only four designations 
can occur, viz. : AA, AB, BA, BB. This proves the proposition, 
and gives one mode of colouring the original map. For the erasure 
of joining lines (such as were originally introduced to divide the 
whole into three-sided compartments) does not necessitate any change 
of lettering. 
This mode of treating the question shows incidentally that in a 
map where only three boundaries meet at each point, the boundaries 
