Mr. J. C. Maxwell on Hills and Dales. 425 



stationary points the equilibrium is stable in some directions and 

 unstable in others. 



On Lines of Slope. 



Lines drawn so as to be everywhere at right angles to the con- 

 tour-lines are called lines of slope. At every point of such a line 

 there is an upward and a downward direction. If we follow the 

 upward direction we shall in general reach a summit, and if we 

 follow the downward direction we shall in general reach a bot- 

 tom. In particular cases, however, we may reach a pass or a bar. 



On Hills and Dales. 



Hence each point of the earth's surface has a line of slope, 

 which begins at a certain summit and ends in a certain bottom. 

 Districts whose lines of slope run to the same bottom are called 

 Basins or Dales. Those whose lines of slope come from the same 

 summit may be called, for want of a better name, Hills. 



Hence the whole earth may be naturally divided into Basins 

 or Dales, and also, by an independent division, into hills, each 

 point of the surface belonging to a certain dale and also to a 

 certain hill. 



On Watersheds and Watercourses. 



Dales are divided from each other by Watersheds, and Hills 

 by Watercourses. 



To draw these lines, begin at a pass or a bar. Here the 

 ground is level, so that we cannot begin to draw a line of slope ; 

 but if we draw a very small closed curve round this point, it will 

 have highest and lowest points, the number of maxima being 

 equal to the number of minima, and each one more than the 

 index number of the pass or bar. From each maximum point 

 draw a line of slope upwards till it reaches a summit. This 

 will be a line of watershed. From each minimum point draw a 

 line of slope downwards till it reaches a bottom. This will be a 

 line of Watercourse. Lines of Watershed are the only lines of 

 slope which do not reach a bottom, and lines of Watercourse are 

 the only lines of slope which do not reach a summit. All other 

 lines of slope diverge from some summit and converge to some 

 bottom, remaining throughout their course in the district be- 

 longing to that summit and that bottom, which is bounded by 

 two watersheds and two watercourses. 



In the pure theory of surfaces there is no method of determi- 

 ning a line of watershed or of watercourse, except by first find- 

 ing a pass or a bar and drawing the line of slope from that 

 point. In nature, water actually trickles down the lines of slope, 

 which generally converge towards the mathematical watercourses, 



Phil. Mag. S. 4. Vol. 40. No. 269. Dec. 1870. 2 F 



