so that in steady motion
| dH | + ∇2ψ | dψ | = 0, | dH | + ∇2ψ | dψ | = 0, | dH | + ∇2ψ = 0, |
| dx | dx | dy | dy | dψ |
(4)
and ∇2ψ must be a function of ψ.
If the motion ia irrotational,
| u = − | dφ | = − | dψ | , v = − | dφ | = | dψ | , |
| dx | dy | dy | dx |
(5)
so that ψ and φ are conjugate functions of x and y,
φ + ψi = ƒ(x + yi), ∇2ψ = 0, ∇2φ = 0;
(6)