_a_, _b_, _c_, _d_, and _e_, successive positions of the fall; _r_, rapid to which the fall is reduced. Draw diagram showing migration of fall in strata dipping _downstream_. Under what conditions of inclination of the strata will a fall migrate the farthest and have the longest life? Under what conditions will it migrate the least distance and soonest be destroyed?] =Waterfalls and rapids.= Before the bed of a stream is reduced to grade it may be broken by abrupt descents which give rise to waterfalls and rapids. Such breaks in a river's bed may belong to the initial surface over which it began its course; still more commonly are they developed in the rock mass through which it is cutting its valley. Thus, wherever a stream leaves harder rocks to flow over softer ones the latter are quickly worn below the level of the former, and a sharp change in slope, with a waterfall or rapid, results. At time of flood young tributaries with steeper courses than that of the trunk stream may bring down stones and finer waste, which the gentler current cannot move along, and throw them as a dam across its way. The rapids thus formed are also ephemeral, for as the gradient of the tributaries is lowered the main stream becomes able to handle the smaller and finer load which they discharge. A rare class of falls is produced where the minor tributaries of a young river are not able to keep pace with their master stream in the erosion of their beds because of their smaller volume, and thus join it by plunging over the side of its gorge. But as the river approaches grade and slackens its down cutting, the tributaries sooner or later overtake it, and effacing their falls, unite with it on a level. [Illustration: Fig. 58. Maturely Dissected Plateau near Charleston, West Virginia Compare the number of streams in any given number of square miles with the number on an area of the same size in the Red River Valley (Fig. 44). What is the shape of the ridges? Are their summits broad or narrow? Are their crests even or broken by knobs and cols (the depressions on the crest line)? If the latter, how deeply have the cols been worn beneath the summits of the knobs?] Waterfalls and rapids of all kinds are evanescent features of a river's youth. Like lakes they are soon destroyed, and if any long time had already elapsed since their formation they would have been obliterated already.