232 Dr. J. McOowan on Ridge-Lines 



through it as apses in the indicatrix, these lines passing 

 through the apses. 



It must be observed that the points here considered, that is 

 to say points at which the tangent plane is horizontal, are 

 not the only multiple points on the ridge locus. The geo- 

 metrical characteristic of the others will be considered later. 



§ 5. Properties of the Ridge-Lines. 



It has already (§ 2) been pointed out that at points on a 

 ridge-line the lines of curvature touch the slope- and contour- 

 lines ; hence their projections touch also. The projections 

 of the slope- and contour-lines cut orthogonally everywhere, 

 but in general those of the lines of curvature do not : they 

 can only do so when they touch the lines of slope and contour. 

 Thus, for all points on a ridge-line, but nowhere else, the pro- 

 jections of the lines of curvature cut orthogonally; or, in other 

 words, the projection of the ridge-line is the locus of the points 

 of orthogonal intersection of the projections of the lines of 

 curvature. 



This is otherwise obvious analytically, for the differential 

 equation to the projections of the lines of curvature is 



{ (1 + q 2 )s-pqt\df + { (1 f q 2 )r- (1 +p 2 )t}dxdy 



+ {pqr-(l+p 2 )s\dx 2 = 0, . . (12) 



and these cut orthogonally wherever (7) is satisfied, for it 

 makes the sum of the coefficients of dx 2 and dy 2 vanish. It 

 may also be noted that (7) is obtained at once from (12) by 

 the substitution of p and q for dx and dy, that is by imposing 

 the condition that lines of slope and lines of curvature are 

 to touch. 



By transformation of (7) other interesting properties of the 

 ridge-lines may be made immediately obvious. 



Thus it may be written 



da da _ .. m 



. ^-^=°> < 13 ) 



where 



tan 2 cr=p 2 + ? 2 (14) 



By (14) a is the angle of slope, that is the inclination of a 

 line of slope or tangent plane to the base, and (13) shows 

 that at any point on a ridge-line this does not vary along a 

 contour. Thus at all points on a ridge-line the contour-lines 

 touch the lines of a constant slope or isoclinics. The same 

 fact may be otherwise expressed by saying that for each 

 contour-line the slope is a minimum or maximum wherever 

 it is crossed by a ridge-dine. In this sense a ridge-line might 



