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



is tlie case of two mouutaiii summits connected by a ridge or 

 col, the lowest point whereof, or head of the pass, is the knot on 

 the outloop contour line through this point. And in like man- 

 ner, that for an inloop curve the lune is an elevation, the inner 

 loop a depression ; and that the outer loop, considered as a por- 

 tion of the contour line, is higher than the consecutive exterior 

 contour line. This is the case of a lake having an outlet; if the 

 lake were dry, the passage up stream into the bed of it would be 

 over a ridge, col, or barrier, the lowest point whereof, or point of 

 outlet for the water of the lake, is the knot on the inloop con- 

 tour line passing through this point, the shore of the lake being 

 of course the inner loop of this contour line, and the waters 

 being retained by means of the raised ground within the lune 

 between the two loops of the contour line. 



The slope lines cut at right angles the contour lines ; and this 

 property applies also to the projections of the two systems of 

 lines ; so that the two systems of lines delineated in piano inter- 

 sect at right angles. Consider the contour lines which are closed 

 curves surrounding a given summit or immit ; the exterior con- 

 tour line is intersected at each of its points by a slope line ; and 

 all these slope lines must, it is clear, intersect all the interior 

 contour lines, and ultimately unite at the interior summit or 

 immit. In order to see more distinctly the form of the system 

 of slope lines, it is to be noticed that, if (as is in general the 

 case) the indicatrix at the summit or immit be an ellipse, the 

 contour lines in the immediate neighbourhood thereof will be a 

 system of similar and similarly situated concentric ellipses, the 

 major and minor axes whereof correspond respectively with the 

 directions of least and greatest curvature; the equation of any 

 orthogonal trajectory of the ellipses, if a, b are the semiaxes, 

 major and minor, of any one of them, is 2/*^ = Ca-"'; and unless 

 C = CO , the curve represented by this equation touches the axis 

 oix, which is the direction of least curvature ; if however C = oo , 

 then the equation becomes .r=0, and the curve touches the axis 

 of y, which is the direction of greatest curvature. Hence in 

 general at a summit or immit the slope curves all, except one 

 (which is a limiting case) touch the line which is the direc- 

 tion of least curvature. The only exception is when the summit 

 or immit is an umbilicus — the indicatrix is then a circle; the 

 contour lines in the immediate neighbourhood of this point are 

 concentric circles, and the slope lines pass in all directions through 

 the summit or immit. 



The indicatrix at a knot is in general a hyperbola, and conse- 

 quently the contour lines in the neighbourhood of a knot are 

 similar and similarly situated concentric hyperbolas ; and if a, b 

 are the semiaxes of one of these hyperbolas, the equation of an 



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