G dQ θ (z + y1 − y0).
Putting pa for atmospheric pressure, the whole pressure per unit of area at a0 is Gy0 + pa, and that at a1 is −(Gy1 + pa). The work of these pressures is
G (y0 + pa/G − y1 − pa/G) dQ θ = G (y0 − y1) dQ θ.
Adding this to the work of gravity, the whole work is GzdQθ; or, for the whole cross section,
GzQθ.
(2)
Work expended in Overcoming the Friction of the Stream Bed.—Let A′B′, A″B″ be two cross sections at distances s and s + ds from A0B0. Between these sections the velocity may be treated as uniform, because by hypothesis the changes of velocity from section to section are gradual. Hence, to this short length of stream the equation for uniform motion is applicable. But in that case the work in overcoming the friction of the stream bed between A′B′ and A″B″ is
GQθζ (u2 / 2g) (χ / Ω) ds,
where u, χ, Ω are the mean velocity, wetted perimeter, and section at A′B′. Hence the whole work lost in friction from A0B0 to A1B1 will be
GQθ ∫10 ζ (u2 / 2g) (χ / Ω) ds.