Now imagine a small cylinder of section ω 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 ωds and its weight Gω ds. Hence, taking the components of the forces parallel to PQ—
ω dp = Gv2ω ds/gρ − Gω cos φ ds,
where ρ is the radius of curvature of the stream line at Q. Consequently, introducing these values in (1),
dH = v2 ds/gρ + v dv/g = (v/g) (v/ρ + dv/ds) ds.
(2)
Currents
§ 34. Rectilinear Current.—Suppose the motion is in parallel straight stream lines (fig. 35) in a vertical plane. Then ρ is infinite, and from eq. (2), § 33,
dH = v dv/g.
Comparing this with (1) we see that
dz + dp/G = 0;