dH = dz + dp/G + vdv/g, (1)
for the increment of energy between Q and P. But
dz = PQ cos [phi] = ds cos [phi];
.: dH = dp/G + v dv/g + ds cos [phi], (1a)
where the last term disappears if the motion is in a horizontal plane.
Now imagine a small cylinder of section [omega] described round PQ as
an axis. This will be in equilibrium under the action of its
centrifugal force, its weight and the pressure on its ends. But its
volume is [omega] ds and its weight G[omega]ds. Hence, taking the
components of the forces parallel to PQ--
[omega]dp = Gv²[omega] ds/g[rho] - G[omega] cos [phi] ds,
where [rho] is the radius of curvature of the stream line at Q.
Consequently, introducing these values in (1),
dH = v² ds/g[rho] + v dv/g = (v/g)(v/[rho] + dv/ds) ds. (2)