Putting the values of du and dz found in (2) and (3) in equation (1),
i ds − dh = −(u2x / gΩ) dh + (χ / Ω) ζ (u2 / 2g) ds.
dh/ds = {i − (χ/Ω) ζ (u2/2g)} / {1 − (u2/g) (x/Ω)}.
(4)
Further Restriction to the Case of a Stream of Rectangular Section and of Indefinite Width.—The equation might be discussed in the form just given, but it becomes a little simpler if restricted in the way just stated. For, if the stream is rectangular, χh = Ω, and if χ is large compared with h, Ω/χ = xh/x = h nearly. Then equation (4) becomes
dh/ds = i (1 − ζu2 / 2gih) / (1 − u2/gh).
(5)
§ 117. General Indications as to the Form of Water Surface furnished by Equation (5).—Let A0A1 (fig. 121) be the water surface, B0B1 the bed in a longitudinal section of the stream, and ab any section at a distance s from B0, the depth ab being h. Suppose B0B1, B0A0 taken as rectangular coordinate axes, then dh/ds is the trigonometric tangent of the angle which the surface of the stream at a makes with the axis B0B1. This tangent dh/ds will be positive, if the stream is increasing in depth in the direction B0B1; negative, if the stream is diminishing in depth from B0 towards B1. If dh/ds = 0, the surface of the stream is parallel to the bed, as in cases of uniform motion. But from equation (4)
dh/ds = 0, if i − (χ/Ω) ζ (u2/2g) = 0;
∴ ζ (u2/2g) = (Ω/χ) i = mi,