The composition of the stream system of a drainage basin can be expressed quantitatively in terms of stream order, drainage density, bifurcation ratio, and stream-length ratio. Stream orders are so chosen that the fingertip or unbranched tributaries are of the 1st order; streams which receive 1st order tributaries, but these only, are of the 2d order; third order streams receive 2d or 1st and 2d order tributaries, and so on, until, finally, the main stream is of the highest order and characterizes the order of the drainage basin. Two fundamental laws connect the numbers and lengths of streams of different orders in a drainage basin: The infiltration theory of surface runoff is based on two fundamental concepts: For a given terrain there is a minimum length x c of overland flow required to produce sufficient runoff volume to initiate erosion. The critical length x c depends on surface slope, runoff intensity, infiltration-capacity, and resistivity of the soil to erosion. This is the most important single factor involved in erosion phenomena and, in particular, in connection with the development of stream systems and their drainage basins by aqueous erosion. The erosive force and the rate at which erosion can take place at a distance x from the watershed line is directly proportional to the runoff intensity, in inches per hour, the distance x , a function of the slope angle, and a proportionality factor K e , which represents the quantity of material which can be torn loose and eroded per unit of time and surface area, with unit runoff intensity, slope, and terrain. The rate of erosion is the quantity of material actually removed from the soil surface per unit of time and area, and this may be governed by either the transporting power of overland flow or the actual rate of erosion, whichever is smaller. If the quantity of material torn loose and carried in suspension in overland flow exceeds the quantity which can be transported, deposition or sedimentation on the soil surface will take place. On newly exposed terrain, resulting, for example, from the recession of a coast line, sheet erosion occurs first where the distance from the watershed line to the coast line first exceeds the critical length x c and sheet erosion spreads laterally as the width of the exposed terrain increases. Erosion of such a newly exposed plane surface initially develops a series of shallow, close-spaced, shoestring gullies or rill channels. The rills flow parallel with or are consequent on the original slope. As a result of various causes, the divides between adjacent rill channels are broken down locally, and the flow in the shallower rill channels more remote from the initial rill is diverted into deeper rills more closely adjacent thereto, and a new system of rill channels is developed having a direction of flow at an angle to the initial rill channels and producing a resultant slope toward the initial rill. This is called cross-grading. With progressive exposure of new terrain, streams develop first at points where the length of overland flow first exceeds the critical length x c , and streams starting at these points generally become the primary or highest-order streams of the ultimate drainage basins. The development of a rilled surface on each side of the main stream, followed by cross-grading, creates lateral slopes toward the main stream, and on these slopes tributary streams develop, usually one on either side, at points where the length of overland flow in the new resultant slope direction first exceeds the critical length x c . Cross-grading and recross-grading of a given portion of the area will continue, accompanied in each case by the development of a new order of tributary streams, until finally the length of overland flow within the remaining areas is everywhere less than the critical length x c . These processes fully account for the geometric-series laws of stream numbers and stream lengths. A belt of no erosion exist