From the Proceedings of the Royal Society of Edinburgh, No. 106, p. 501, it appears the colouring of maps is receiving attention. This note bears chiefly upon the history of the matter.

Some thirty years ago, when I was attending Professor De Morgan's class, my brother, Francis Guthrie, who had recently ceased to attend them (and who is now professor of mathematics at the South African University, Cape Town), showed me the fact that the greatest necessary number of colours to be used in colouring a map so as to avoid identity of colour in lineally contiguous districts is four. I should not be justified, after this lapse of time, in trying to give his proof, but the critical diagram was as in the margin. With my brother's permission I submitted the theorem to Professor De Morgan, who expressed himself very pleased with it; accepted it as new; and, as I am informed by those who subsequently attended his classes, was in the habit of acknowledging whence he had got his information.

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 I have therefore withdrawn my former paper, as in great part superseded by the present one.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

It was shown some time ago by Dr Ferguson, and more recently by Professor Chrystal and Mr Preece, that a fine wire attached to a mechanical telephone can act very well as a receiver in a telephonic circuit, provided a make and break, or some form of microphone transmitter, be employed. None of these experimenters, however, have said anything about the action of such a wire as a transmitter. Being struck by the convertibility, in general, of all forms of telephone receivers into transmitters, and vice versa, it occurred to me to try how far this wire telephone, as it has been called, could be made to act as a transmitter to an ordinary Bell telephone as receiver. I was much interested to find that it could act in that capacity wonderfully well, as thereby a new element of some importance is introduced into the discussion of the real cause or causes of the action of the wire telephone whether as receiver or as transmitter.

This note is a continuation of the paper on the “Preparation and Properties of Pure Graphitoid and Adamantine Boron,” by Dr R. M. Morrison and myself, published in the Transactions of this Society, vol. xxviii. p. 689, and its object is to correct a mistake which we have made in giving the properties of this substance. We say—

1st, It is not oxidised by air at a white heat, even superficially.

2d, It does not alloy with platinum at a white heat.

I have again prepared a quantity of this substance and examined its properties under Professor Wohler, and I find that the above statements are wrong. A very fine film of oxide does form on the surface of this substance when heated on platinum foil over the blowpipe, and this film, although very thin and difficult to observe, is sufficient to prevent all further action of the air, and also to prevent its combining with the platinum. This misled us when we previously examined it. If, however, a quantity of it be placed on a piece of platinum foil, and the foil folded over it and pressed down so as to exclude all the air, then on heating it intensely before the blow-pipe the boron at once combines with the platinum and perforates it.