r sin nθ = na.
(11)
A single vortex in a circular cylinder of radius a at a distance c from the centre will move with the velocity due to an equal opposite image at a distance a2/c, and so describe a circle with velocity
mc/(a2 − c2) in the periodic time 2π (a2 − c2)/m.
(12)
Conjugate functions can be employed also for the motion of liquid in a thin sheet between two concentric spherical surfaces; the components of velocity along the meridian and parallel in colatitude θ and longitude λ can be written
| dφ | = | 1 | dψ | , | 1 | dψ | = − | dψ | , | ||
| dθ | sin θ | dλ | sin θ | dλ | dθ |
(13)
and then
φ + ψi = F (tan ½θ·eλi).