=The courses of valleys.=—River valleys are rarely straight. To understand why they are crooked it is only necessary to understand the methods by which they grow. In so far as a river valley is a gully grown big, that is, in so far as its length is the result of head erosion, its course was determined by the course of the antecedent gully. If in the case shown in Fig. 59 the slope of the surface above the head of the gully is uniform, its material homogeneous, and the rainfall everywhere equal, more water will come into the gully from the direction _a_ than from any other. In this case there would be more wear in the direct line of its extension than elsewhere, and the head would advance in a straight line. But if there be inequalities of slope about the head of a gully at any stage of its development more water may come in from some direction other than that in the direct line of its extension. In Fig. 59, for example, more water may enter from the direction of _b_ than from that of _a_. Since most wear is likely to be affected along the line of greatest inflow, the head of the gully will be turned in that direction (Fig. 60). Started in this course it will continue in the new direction so long as erosion in this line is greater than that elsewhere; but whenever the configuration of the surface causes more water to enter the head of the gully from some direction other than that in which it is headed, the line of axial growth is again changed, as toward _c_, Fig. 60. Since new land surfaces are probably more or less undulatory, crookedness should be the rule among valleys developed from gullies by head erosion. Streams and valleys the courses of which are determined by the original slope of the land are said to be _consequent_. [Illustration: +Fig.+ 61.—Diagram illustrating the development of two equal gullies from the head of one.] Inequalities of material, leading to unequal rates of erosion, effect the same result, in the absence of inequalities of slope. If at any stage of a valley’s development erosion were equal in two directions at its head, and at the same time greater than at points between, two gullies would result (Fig. 61) diverging from the point in question. In the case of a valley developed by overflow from a lake its course is determined by the lowest line of flow to which the water has access. If this line be straight the valley will be straight; if it is crooked, as it generally is, the valley is crooked also. =The development of tributaries.=—Thus far valleys leading immediately to the sea have been considered, and no account taken of tributaries. As a matter of fact most considerable valleys have numerous tributaries. It is now in order to inquire into their mode of development.