of Edinburgh, Session 1876-80. 
729 
10. Kemarks on the previous Communication. By Prof. Tait. 
{Abstract.) 
In a paper read to the Society on 15th March last (ante, p. 501), 
I gave a series of proofs of the theorem that four colours suffice for 
a map. All of these were long, and I felt that, while more than 
sufficient to prove the truth of the theorem, they gave little insight 
into its real nature and hearings. A somewhat similar remark may, 
I think, be made about Mr Kempe’s proof. 
But a remark incidentally made in the abstract of my former 
paper has led me to a totally different mode of attacking the 
question, which puts its nature in a clearer light. I have therefore 
withdrawn my former paper, as in great part superseded by the 
present one. 
The remark referred to is to the effect that, if an even number of 
points he joined, so that three (and only three) lines meet in each, 
these lines may be coloured with three colours only, so that no 
two conterminous lines shall have the same colour. (When an odd 
number of the points forms a group, connected by one line only 
with the rest, the theorem is not true.) 
This follows immediately from the main theorem when it is 
applied to a map in which the boundaries meet in threes (and the 
excepted case cannot then present itself). Por we have only to 
colour such a map with the colours A, B, C, D. Then if the 
common boundaries of A and B, as also of C and D, be coloured 
a; those of A and C, and of B and D, (3 ; and those of A and D, 
and of B and C, y , it is clear that the three boundaries which meet 
in any one point will have the three colours a, (3 , y. 
The proof of the elementary theorem is given easily by induc- 
tion ; and then the proof that four colours suffice for a map fol- 
lows almost immediately from the theorem, by an inversion of the 
demonstration just given. 
We escape the excepted case by taking the points as the summits 
of a polyhedron, all of which are trihedral ; and when the figure is 
a pentagonal dodecahedron the theorem leads to Hamilton’s Icosian 
Game. 
