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194 rested hitherto, as far as I know, on the experience I have mentioned, and on the statement of Professor De Morgan, that the fact was no doubt true. Whether that statement was one merely of belief, or whether Professor De Morgan, or any one else, ever gave a proof of it, or a way of colouring any given map, is, I believe, unknown; at all events, no answer has been given to the query to that effect put by Professor Cayley to the London Mathematical Society on June 13th, 1878, and subsequently, in a short communication to the Proceedings of the Royal Geographical Society, Vol. I, p. 259, Professor Cayley, while indicating wherein the difficulty of the question consisted, states that he had not then obtained a solution. Some inkling of the nature of the difficulty of the question, unless its weak point be discovered and attacked, may be derived from the fact that a very small alteration in one part of a map may render it necessary to recolour it throughout. After a somewhat arduous search, I have succeeded, suddenly, as might be expected, in hitting upon the weak point, which proved an easy one to attack. The result is, that the experience of the map-makers has not deceived them, the maps they had to deal with, viz: those drawn on simply connected surfaces, can, in every case, be painted with four colours. How this can be done I will endeavourat the request of the Editor-in-Chief-to explain.

Suppose that we have the surface divided into districts in any way which admits of the districts being coloured with four colours, viz: blue, yellow, red, and green; and suppose that the districts are so coloured. Now if we direct our attention to those districts which are coloured red and green, we shall find that they form one or more detached regions, i. e. regions which have no boundary in common, though possibly they may meet at a point or points. These regions will be surrounded by and surround other regions composed of blue and yellow districts, the two sets of regions making up the whole surface. It will readily be seen that we can interchange the colours of the districts in one or more of the red and green regions without doing so in any others, and the map will still be properly coloured. The same remarks apply to the regions composed of districts of any other pair of colours. Now if a region comuposed of districts of any pair of colours, say red and green as before, be of either of the forms shown in Figures 3 and 4, it will separate the surface into two parts, so that we may be quite certain that no yellow or blue districts in one part can belong, to the same yellow and blue region as any yellow or blue district in the other part. Thus any specified blue district, for example, in one part can, by an interchange of the colours in the yellow and