Let AB, CD (fig. 34) be two consecutive stream lines, at present assumed to be in a vertical plane, and PQ a normal to these lines making an angle φ with the vertical. Let P, Q be two particles moving along these lines at a distance PQ = ds, and let z be the height of Q above the horizontal plane with reference to which the energy is measured, v its velocity, and p its pressure. Then, if H is the total energy at Q per unit of weight of fluid,
H = z + p/G + v2/2g.
Differentiating, we get
dH = dz + dp/G + v dv/g,
(1)
for the increment of energy between Q and P. But
dz = PQ cos φ = ds cos φ;
∴ dH = dp/G + v dv/g + ds cos φ,
(1a)
where the last term disappears if the motion is in a horizontal plane.