I conceive that the explanation of these phenomena is as follows: The heads of the secondary drainage lines laid down in the diagram are in nature tolerably definite points. The water which during rain converges at one of these points is there abruptly concentrated in volume. Above the point it is a sheet, or at least is divided into many rills. Below it, it is a single stream with greatly increased power of transportation and corrasion. The principle of equal action gives to the concentrated stream a less declivity than to the diffused sheet, and—what is especially important—it tends to produce an equal grade in all directions upward from the point of convergence. The converging surface becomes hopper-shaped or funnel-shaped; and as the point of convergence is lowered by corrasion, the walls of the funnel are eaten back equally in all directions—except of course the direction of the stream. The influence of the stream in stimulating erosion above its head is thus extended radially and equally through an arc of 180°, of which the center is at the point of convergence. Where two streams head near each other, the influence of each tends to pare away the divide between them, and by paring to carry it farther back. The position of the divide is determined by the two influences combined and represents the line of equilibrium between them. The influences being radial from the points of convergence, the line of equilibrium is tangential, and is consequently at right angles to a line connecting the two points. Thus, for example, if _a_, _b_, and _c_ (Figure 58) are the points of convergence at the heads of three drainage lines, the divide line _ed_ is at right angles to a line connecting _a_ and _b_, and the divides _fd_ and _gd_ are similarly determined. The point _d_ is simultaneously determined by the intersection of the three divide lines. [Illustration: FIG. 58.—Ground plan of a Bad-land Ridge, showing its relation to Waterways. The smooth lines represent Divides. FIG. 59.—Profile of the same ridge. ] Furthermore, since that point of the line _ed_ which lies directly between _a_ and _b_ is nearest to those points, it is the point of the divide most subject to the erosive influences which radiate from _a_ and _b_, and it is consequently degraded lower than the contiguous portions of the divide. The points _d_ and _e_ are less reduced; and _d_, which can be shown by similar reasoning to stand higher than the adjacent portion of either of the three ridges which there unite, is a local maximum.