Mr. A. Cayley on Contour and Slope Lines. 267 



orthogonal trajectory is x'^\/^-=zQ ; and when this passes through 

 the knot, C = j and therefore either .r = or else ?/ = 0; there 

 are consequently through the knot only two slope lines, which 

 bisect the angles made by the two branches of the contour line 

 and intersect each other at right angles. The slope lines through 

 a knot may be termed ridge and course lines : and for one of 

 these — the ridge line — the knot is a point of minimum elevation ; 

 for the other of them — the course line — the knot is a point 

 of maximum elevation. But this requires some further develop- 

 ment. To fix the ideas, consider the case where the contour 

 line is an outloop curve, the loops being each of them elevations. 

 The slope line through the knot, and which lies within the two 

 loops, would be, according to the definition, a ridge line. Suppose 

 that the contour lines within one of the loops are closed curves 

 surrounding a summit, the ridge line will, it is clear, cut all 

 these curves and ultimately ari'ive at the summit. But if the 

 contour lines within the loop are not all of them closed curves ; 

 if, for instance, they are first closed curves, then an outloop 

 curve, and within each of the loops of this, closed curves sur- 

 rounding a summit, then it may happen that the above-mentioned 

 ridge line will pass through the knot of the inner outloop curve : 

 and with respect to this knot, it will be, not a ridge line, but a 

 course line ; so that the slope line in question cannot be spoken 

 oi simpliciter either as a ridge line or as a course line, but it is the 

 one or the other quoad the knot in reference to which it is con- 

 sidered ; and, considered by itself, it can only be spoken of as a 

 ridge- or- course line. The case just referred to is, however, an 

 exceptional one ; in general the slope line in question would not 

 pass through the knot of the inner outloop curve, but would cut 

 one of the loops of this curve, and then cutting all the contour 

 lines within such loop, arrive at last at the summit within such 

 loop. And when the ridge line has once arrived at a summit, 

 there is little meaning in continuing it further, and it may be 

 considered as ending there ; iu fact there are through the sum- 

 mit an infinity of slope lines, all of them (except in the case 

 where the summit is an umbilicus) coincident in direction with 

 the ridge line, and consequently the ridge line may, without 

 graphical discontinuity, be considered as proceeding along any 

 one of these lines indifferently ; and although, when the sui'face 

 is a geometrical one capable of being represented by an equation, 

 there would be geometrically one of these slope lines which could 

 be identified as the continuation of the ridge line, there would be 

 no advantage iu making this identification. Hence it may be con- 

 sidered that in general a ridge line passes from summit to sum- 

 mit, through a single intervening knot which is a point of 

 minimum elevation on the ridge line ; and in like manner, that 



