236 On Ridge-Lines and Lines connected with them, 



§ 8. General Considerations. 



Certain general inferences from equation (7) deserve notice. 

 If F(^) be substituted in it for z, it is only altered by the 

 introduction of a factor (the cube of F' (,?)). Thus if two 

 surfaces have the same lines for the projection of their con- 

 tours, the projections of their ridge -lines will also be the 

 same, except that certain contour-lines (given by F' (z) = 0) 

 may be ridge-lines for one of the surfaces only, ridge-lines 

 all along which the surface touches horizontal tangent planes. 



Again, a surface may have no definite ridge-lines, or, in 

 other words, all points on the surface may satisfy (7) : in fact 

 (7) may be regarded as the partial differential equation of 

 such surfaces. In the form (13) the solution of this equation 

 is obvious : thus, surfaces whose slope at any point depends 

 only on the height of the point above the base have no definite 

 ridge-lines, or, as it may be perhaps better put, every line of 

 slope and every contour-line on such a surface is a ridge-line. 

 Surfaces of revolution obviously belong to this class. 



The conditions that a point on a surface may be an umbilic 

 are 



lTp~ 2 = ^ = T+q 72 ' ( 22 ) 



for these make dyjdx indeterminate in (12). These ratios 

 make (7) vanish identically : hence all the umbilics of any 

 surface lie on ridge-lines. This is otherwise evident from the 

 consideration that in the neighbourhood of an umbilic the 

 lines of curvature have all directions. Again, (7) is identically 

 satisfied at points where r = s = t = ; that is to say, at points 

 where the surface has no curvature. All such points there- 

 fore lie on the ridge-lines. 



Again, it is geometrically obvious — or may be proved as 

 below— that all singular points at which there is a tangent 

 cone lie on ridge-lines. The three classes of points here 

 shown to lie on the ridge-lines are peculiar to the surface : 

 they have no relation to the base. This leads to the interest- 

 ing result that while in general the ridge-lines on a surface 

 will continuously vary when it is moved relatively to its base, 

 these points will be fixed points through which they will 

 continue to pass. 



As the equation to a surface is in general most simply 

 expressed in the form 



4>(x, y, z)=c, (23) 



it is desirable before concluding to give the corresponding 

 equation determining the ridge-lines. Still taking 2=0 as 



