Hydraulic mean depth = ab = a √ (kl/cv).
But, by the ordinary formula for the flow of rivers, mi = ζv2;
∴ i = ζv2 / m = (ζv5/2 / a) √ (c / kl).
But i is the tangent of the angle which the curve at C makes with the axis of X, and is therefore = dy/dx. Also, as the slope is small, l = AC = AD = x nearly.
∴ dy/dx = (ζv5/2 / a) √ (c / kx);
and, remembering that v is constant,
y = (2ζv5/2 / a) √ (cx / k);
or
y2 = constant × x;
so that the curve is a common parabola, of which the axis is horizontal and the vertex at the source. This may be considered an ideal longitudinal section, to which actual rivers approximate more or less, with exceptions due to the varying hardness of their beds, and the irregular manner in which their volume increases.