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L. On Hills and Dales. 

 By J. Clerk Maxwell, LL.D., F.R.S. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



I FIND that in the greater part of the substance of the fol- 

 lowing paper I have been anticipated by Professor Cayley, 

 in a memoir " On Contour and Slope Lines," published in the 

 Philosophical Magazine in 1859 (S. 4. vol, xviii. p. 264). An 

 exact knowledge of the first elements of physical geography, 

 however, is so important, and loose notions on the subject are 

 so prevalent, that I have no hesitation in sending you what you, 

 I hope, will have no scruple in rejecting if you think it superfluous 

 after what has been done by Professor Cayley. 

 I am, Gentlemen, 



Your obedient 'Servant, 



J. Clerk Maxwell. 

 Glenlair, Dalbeattie, 

 October 12, 1870. 



1. On Contour-lines and Measurement of Heights. 



The results of the survey of the surface of a country are most 

 conveniently exhibited by means of a map on which are traced 

 contour- lines, each contour-line representing the intersection of 

 a level surface with the surface of the earth, and being distin- 

 guished by a numeral which indicates the level surface to which 

 it belongs. 



When the extent of country surveyed is small, the contour- 

 lines are defined with sufficient accuracy by the number of feet 

 above the mean level of the sea ; but when the survey is so ex- 

 tensive that the variation of the force of gravity must be taken 

 into account, we must adopt a new definition of the height of a 

 place in order to be mathematically accurate. If we could de- 

 termine the exact form of the surface of equilibrium of the sea, 

 so as to know its position in the interior of a continent, we might 

 draw a normal to this surface from the top of a mountain, and 

 call this the height of the mountain. This would be perfectly 

 definite in the case when the surface of equilibrium is everywhere 

 convex ; but the lines of equal height would not be level surfaces. 



Level surfaces are surfaces of equilibrium, and they are not 

 equidistant. The only thing which is constant is the amount of 

 work required to rise from one to another. Hence the only con- 

 sistent definition of a level surface is obtained by assuming a 

 standard station, say, at the mean level of the sea at a particular 



