228 Dr. J. McCowan on Ridge-Lines 



called summits and immits respectively : all lines of slope run 

 from summits to immits, and obviously cannot cross each other 

 at any intermediate point, though a line may branch at any 

 point where there is a horizontal tangent plane. At summits 

 and immits the surface is synclastic and the contours are 

 reduced to points ; but a horizontal plane may touch the sur- 

 face at a point, called a col, where the surface is anticlastic, 

 and where consequently two branches of a contour-line will in 

 general intersect, the point of contact being a double point of 

 the curve of section. Other varieties of contact need not be 

 specially noted here as their occurrence is very exceptional. In 

 the case of a land surface the summits are the tops of the hills, 

 and the immits the bottoms of lakes or basins, while the cols 

 are such points as the heads of passes between hills or places 

 where streams flow out from lakes. The two lines of slope 

 from a col to the two adjacent summits form together what is 

 called a watershed, and the line of slope from a col to an 

 immit is called a watercourse"*. 



§ 2. Definition of Ridge-Lines. 



The section of a surface by a plane parallel and very near 

 to the tangent plane at any point, or, as it is generally called, 

 the indicatrix for that point, is in general a conic. The 

 directions of the axes of this conic, depending only on the 

 form of the surface and not at all on its relation to the base, 

 are independent of the directions of the slope- and contour- 

 lines, and therefore the surface in the immediate neighbour- 

 hood of any point is in general unsymmetrical about the line 

 of slope through the point. There are, however, points on 

 the surface at which the directions of the principal axes of 

 the indicatrix coincide with the directions of the slope- and 

 contour-lines, and the locus of such points is a line which I 

 propose to call a ridge-line. The ridge-line, then, is defined 

 by this property: — A surface, in the immediate neighbourhood 

 of any point on a ridge-line, is symmetrical about the line of 

 slope through that point. 



The topographical significance of the line is to be noted. 

 The features of a hilly country which are, perhaps, most cha- 

 racteristic are the ridges and spurs of the hills and the glens 

 or valleys between them. The general trend of these features 

 is given by the ridge-lines, each ridge-line being, so to speak, 

 the central or axial line, the line of (infinitesimal) symmetry 

 of a ridge or valley — the valleys may be regarded as negative 



* For more detailed treatment of the matters of this section reference 

 may be made to the papers of Cayley and Maxwell, he. cit. ante. 



