and Lines connected with them. 233 



be called a line of minimum or maximum slope, and might 

 have been defined by this property had not the definition 

 chosen appeared the more fundamental. 

 Again, (7) may be written 



PTx+IJy^' (15) 



where 



tan 6= —p/g 9 (16) 



and 6 is therefore the inclination of the projection of the line 

 of slope to the axis of y. Thus (15) shows that at points on 

 a ridge-line the projections of the lines of slope have no cur- 

 vature, or, in other words, the projection of the ridge-line 

 passes through the points of inflexion of the projections of 

 the lines of slope. 



§ 6. The Ridge-Lines as particular members of a Family 



of Lines. 



The equations (13) and (15) suggest at once another point 

 of view from which the ridge-lines may be regarded, namely, 

 as members of the family of curves whose parameter y is 

 determined by one of the equivalent equations 



(p 2 -q*)s-pq{r-t) = {p 2 + q 2 )*/ 2 y, . . . (17) 

 ^q=Y> (18) 



S-* ^ 



where dC and dS are elements of contour- and slope-line 

 projections respectively. The parameter y is by (19) the 

 curvature of the projection of the line of slope at any point, 

 or, by (18), it may be interpreted as the rate at which at any 

 point the slope -fine turns round the contour-line as an axis 

 per unit distance of advance along the contour. In passing, 

 the interesting theorem thus given may be noted, namely, 

 that the rate of twist of a slope-line about a contour-line at 

 any point on a surface is equal to the curvature of the pro- 

 jection of the slope-line at the same point. 



The ridge-line is the curve for which the parameter 7 is 

 zero, a value which obviously reduces the equations (17), (18), 

 and (19) to the forms (7), (13), and (15) respectively. The 

 curves for the general case, 7= constant, have not much 

 special interest, but the consideration of 7 leads to a simple 

 geometrical characterization of those multiple points on the 

 ridge-lines which were left over in § 4. 



