If the surface was so changed as to allow of the development of a valley (p. 63) the same principles would be applicable. As an active stream passes from a hard layer to one less resistant, the greater wear on the latter gives origin to rapids. At first the rapids would be slight (_a_, Fig. 112), but would become more considerable (_b_) as time and erosion go on. When the bed of the rapids becomes sufficiently steep, the rapids become _falls_[49] (_cd_). When the water falls rather than flows over the rock surface below the hard layer, erosion assumes a new phase. The hard layer is then undermined, and the undermining causes the falls to recede. This phase of erosion is sometimes called _sapping_. [Illustration: +Fig.+ 111.—Diagram representing a horizontal layer of hard rock in an island, and its effects on erosion.] [Illustration: +Fig.+ 112.—Diagram illustrating the development of a fall where the hard layer dips gently up-stream.] [Illustration: +Fig.+ 113.—Diagram illustrating the conditions which exist at Niagara Falls. (Gilbert.)] If the hard layer which occasions a fall dips up-stream (Fig. 112), its outcrop in the stream’s bed becomes lower as the fall recedes (_e_). When it has become so low that the water passing over it no longer reacts effectively against the less resistant material beneath (_f_), sapping ceases, and the point of greatest erosion may be shifted from the soft material beneath the fall to the hard layer itself. The actual rate of erosion at this point may be no greater than before, though the relative rate is. Under these circumstances the vertical edge of the hard layer will presently be converted into an incline (_f_), and as this takes place the fall becomes rapids. The conversion of the falls into the rapids begins about the time the lower edge of the hard stratum in the channel reaches grade. By continuation of the process which transformed the falls into rapids, the rapids become less rapid, and when the upper edge of the hard layer has been brought to grade, the rapids disappear (_h_, Fig. 112). The history of rapids which succeed falls is the reverse of that which preceded. The later rapids are steepest at the beginning of their history, the earlier at their end. Stated in other terms, rapids are steepest when nearest falls in time. Slight differences in hardness in successive layers often occasion successive falls or rapids (Fig. 114). If the hard layer which occasions the falls be horizontal, instead of dipping up-stream, the general result would be the same; but, other things being equal, the duration of the falls developed under these conditions would be greater, since they must recede farther before becoming rapids.