K = ∫ dp/ρ + V + ½q2 = H
(3)
is constant along a vortex line, and a stream line, the path of a fluid particle, so that the fluid is traversed by a series of H surfaces, each covered by a network of stream lines and vortex lines; and if the motion is irrotational H is a constant throughout the fluid.
Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = 0, ξ = 0; and in steady motion the equations reduce to
dH/dν = 2vζ − 2wη = 2qω sin θ,
(4)
where θ is the angle between the stream line and vortex line; and this holds for their projection on any plane to which dν is drawn perpendicular.
In plane motion (4) reduces to
| dH | = 2qζ = q ( | dQ | + | q | ), |
| dν | dv | r |
(5)