230 Dr. J. McCowan on Ridge-Lines 



The directions of the axis of the conic (4) are, however, 

 given by 



(cos 2 (j>— sin 2 <£)s = cos<£.sin<£ . (r— t) ; . . (6) 



and on eliminating <j> by (5) this gives 



(p 2 -q 2 )s=pq{r-t); (7) 



which must hold at all points on a ridge-line, and is therefore 

 the equation to its projection. Equation (7) maybe regarded 

 as that of a cylinder whose intersection with the surface, given 

 by (1), is the ridge-line. It will, however, be convenient to 

 regard parts of the general locus as separate ridge-lines ; and 

 in general each of the branches which passes through a mul- 

 tiple point, such as a summit or immit (v. infra), will be 

 spoken of as a separate ridge or ridge-line. 



It may be remarked that if the surface is given by an 

 equation such as (1) of the nth. degree, the equation (7) to 

 the projections of the ridge-lines is of degree 3n — 4. 



It is interesting to note, and the fact constitutes a not 

 unimportant claim of the ridge-lines to attention, that whereas 

 in many cases lines of slope and contour-lines cannot be found 

 at all, being determined by differential equations which are in 

 general not integrable in finite terms, the ridge-lines can 

 always be deduced directly from the equation to the surface 

 by mere differentiation. 



§ 4. The Relation of the Summits, Immits, Sfc. to the 

 Ridge-Lines. 



The equation (7) which determines the ridge-lines is 

 satisfied by p=0, <? = 0, and therefore— as is also geometri- 

 cally obvious — all points at which the tangent plane is hori- 

 zontal, that is to say, all summits, immits, cols, &c. lie on the 

 ridge-lines : a result in accordance with the popular notion 

 of a ridg'e, namely, a line running along the summits of a 

 range of hills. 



At such points equation (2), which gives the form of the 

 adjacent surface, reduces, on neglecting terms of the third 

 degree, to 



i=z + i{rP + 28fy + tr?}; .... (8) 



and therefore by (7), the projections of the ridge-lines are 

 given by 



{(r% + sy) 2 -~(sZ + t'n) 2 }s = (rZ + S'n)(s% + tr))(r-t), 



which reduces to 



(rt-s 2 ){s(?-- V 2 )-(r-t)t V }, .... (9) 



