which would be its value if all the particles passing the section had the same velocity u0. Let the kinetic energy be taken at
α (Gθ / 2g) Ω0u03 = α (Gθ / 2g) Qu02,
where α is a corrective factor, the value of which was estimated by J. B. C. J. Bélanger at 1.1.[6] Its precise value is not of great importance.
In a similar way we should obtain for the kinetic energy of A1B1C1D1 the expression
α (Gθ / 2g) Ω1u13 = α (Gθ / 2g) Qu12,
where Ω1, u1 are the section and mean velocity at A1B1, and where a may be taken to have the same value as before without any important error.
Hence the change of kinetic energy in the whole mass A0B0A1B1 in θ seconds is
α (Gθ / 2g) Q (u12 − u02).
(1)
Motive Work of the Weight and Pressures.—Consider a small filament a0a1 which comes in θ seconds to c0c1. The work done by gravity during that movement is the same as if the portion a0c0 were carried to a1c1. Let dQ θ be the volume of a0c0 or a1c1, and y0, y1 the depths of a0, a1 from the surface of the stream. Then the volume dQ θ or G dQ θ pounds falls through a vertical height z + y1 − y0, and the work done by gravity is