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XIX. On Ridge-Lines and Lines connected with them. 

 By J. McCowan, M.A., D.Sc, University College, Dundee*. 



THE topography of mountainous regions, districts of 

 upland and valley, was discussed by Cay ley, in 1859, 

 in a paper "On Contour and Slope Lines "f, and again by 

 Maxwell, in 1870, in a paper " On Hills and Dales" J. So far 

 as contour- and slope-lines are immediately concerned the dis- 

 cussion may therefore be regarded as complete ; but I desire 

 to define and call attention to certain lines connected with 

 these which I shall discuss in some detail in the following 

 paper, and which are especially characteristic of the general 

 configuration of a region of mountain and valley. These 

 lines I have called ridge-lines, but they are not to be con- 

 founded with those to which Cayley gave that name. The 

 word "ridge "as he employs it seems in general usage to have 

 given place to the term " watershed," used in its stead by Max- 

 well ; so perhaps I may be permitted to transfer it to the 

 lines I wish to discuss, as being specially descriptive of them. 

 It may be noted that in general there will be a ridge, as I 

 define it very near to, and in some cases coincident with, the 

 particular line of slope to which Cayley gave the name. 



§ 1. Contour-Lines and Lines of Slope. 



Consider the configuration of a surface, of any form, rela- 

 tively to a plane fixed with respect to it. This plane will be 

 called the base and will be regarded as horizontal, so that 

 planes parallel and perpendicular to it may be described as 

 horizontal and vertical planes respectively. The surface may 

 be, for example, that of any portion of land, and ilie base the 

 sea-level, provided that the part considered is not so large as 

 to require the curvature of the earth's surface to be taken 

 into account. 



The curves in which the surface is intersected by horizontal 

 planes are called contour-lines, or simply contours. Lines on 

 the surface which cut the contours orthogonally are called 

 lines of slope : the inclination at any point of a line of slope to 

 the base is therefore equal to the inclination to the base of the 

 tangent plane at the same point and is a measure of the slope 

 there. Points at a maximum or minimum height above the 

 base at which horizontal tangent planes touch the surface are 



* Cornrnunicated by the Author. Read before the Edinburgh Mathe- 

 matical Society, December 8, 1893. 



t Phil. Mag. [4] vol, xviii. ; or ' Collected Papers/ vol. iv, 

 \ Phil. Mag. |_4| vol. xl. ; or ' Collected Papers,' vol. ii. 



