CH. Ill] GEOMETRICAL RECREATIONS 59 



more than four tints to colour it, and there is no reason to 

 doubt the correctness of the statement that it is not necessary 

 to have more than four colours for any plane map. The 

 number of ways in which such a map can be coloured with four 

 tints has been also considered*, but the results are not suffi- 

 ciently interesting to require mention here. 



I believe that in the corresponding question with solids in 

 space of three dimensions not more than six tints are required 

 to colour the exposed surfaces, but I have never seen any 

 attempt to prove this extension of the problem. 



Physical Configuration of a Country. As I have been 

 alluding to maps, I may here mention that the theory of 

 the representation of the physical configuration of a county 

 by means of lines drawn on a map was discussed by Cayley 

 and Clerk Maxwellf. They showed that a certain relation 

 exists between the number of hills, dales, passes, &c. which 

 can co-exist on the earth or on an island. I proceed to give a 

 summary of their nomenclature and conclusions. 



All places whose heights above the mean sea level are equal 

 are on the same level. The locus of such points on a map is 

 indicated by a contour-line. Roughly speaking, an island is 

 bounded by a contour-line. It is usual to draw the successive 

 contour-lines on a map so that the difference between the 

 heights of any two successive lines is the same, and thus the 

 closer the contour-lines the steeper is the slope, but the 

 heights are measured dynamically by the amount of work to 

 be done to go from one level to the other and not by linear 

 distances. 



A contour-line in general will be a closed curve. This 

 curve may enclose a region of elevation: if two such regions 



* See A. C. Dixon, Messenger of Mathematics, Cambridge, 1902-3, vol. xxxn, 

 pp. 81—83. 



t Cayley on ' Contour and Slope Lines,' Philosophical Magazine, London, 

 October, 1859, series 4, vol. xvm, pp. 264 — 268; Collected Works, vol. rv, 

 pp. 108 — 111. J. Clerk Maxwell on 'Hills and Dales,' Philosophical Magazine, 

 December, 1870, series 4, vol. xl, pp. 421—427; Collected Works, vol. n, 

 pp. 233—240. 



