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 bearings. 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 I have therefore withdrawn my former paper, as in great part superseded by the present one.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 be 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.)