or putting
φ + ψi = w, x + yi = z, w = ƒ(z).
The curves φ = constant and ψ = constant form an orthogonal system; and the interchange of φ and ψ will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.
For instance, in a uniplanar flow, radially inward towards O, the flow across any circle of radius r being the same and denoted by 2πm, the velocity must be m/r, and
φ = m log r, ψ = mθ, φ + ψi = m log reiθ, w = m log z.
(7)
Interchanging these values
ψ = m log r, φ = mθ, ψ + φi = m log reiθ
(8)
gives a state of vortex motion, circulating round Oz, called a straight or columnar vortex.