and Lines connected with them. 229 



or inverted ridges. The ridge-lines run down the bottoms of 

 the glens or along the tops of the ridges and spurs of the 

 hills, or again down the lines where the descent is most 

 precipitous. 



Many of the most interesting properties of the ridge-lines 

 may be derived almost immediately from the definition. For 

 instance, it may be noted that wherever a line of slope or 

 contour is crossed by a ridge-line, the lines of slope or contour 

 touch the lines of curvature of the surface, for the directions 

 of principal curvature are given by the axes of the indicatrix. 

 It is most convenient, however, to proceed at once to the 

 equation to the ridge-lines from which all such properties 

 almost immediately follow. 



§ 3. The Equation to the Ridge-Lines, 



Take the base as the plane s = 0, z being measured verti- 

 cally upwards. Let f , 77, f be the coordinates of a point on 

 the surface near to the point x, y, z also on the surface. 

 Suppose, for the present, the surface to be determined by the 

 equation 



*=/foy) (i) 



Then, employing the usual notation for the partial differential 

 coefficients of z, the surface may, by Taylor's Theorem, be 

 represented in the immediate neighbourhood of the point 

 x, y, z by 



;= +^ + ^ + i{^f + 2^7 ? + ^ 2 }+&c. ... (2) 



The indicatrix for the point x, y, z, that is, the section of the 

 surface by the plane, parallel and very near to the tangent 

 plane, 



r-r=*+j>?+2i», (3) 



will therefore be determined by 



2Z > / =r? + 2sZ v + t v 2 , (4) 



which is the equation to the projection of the indicatrix on 

 the base. It will be convenient in future to use the word 

 " projection " simply, whenever a projection on the base is to 

 be understood. 



At any point on a ridge-line the line of slope touches a 

 principal axis of the indicatrix : the projections of these lines 

 therefore touch also ; and so, if (f> be the angle the projection 

 of that axis of the indicatrix which touches the line of slope 

 makes with the axis of x, 



p/cos<jE> = ^/sm^>. ..... * (5) 



Phil Mag. S. 5. Vol. 37. No. 225. Feb. 1894. R 



