A single vortex will remain at rest, and cause a velocity at any point inversely as the distance from the axis and perpendicular to its direction; analogous to the magnetic field of a straight electric current.
If other vortices are present, any one may be supposed to move with the velocity due to the others, the resultant stream-function being
ψ = Σm log r = log Πrm;
(9)
the path of a vortex is obtained by equating the value of ψ at the vortex to a constant, omitting the rm of the vortex itself.
When the liquid is bounded by a cylindrical surface, the motion of a vortex inside may be determined as due to a series of vortex-images, so arranged as to make the flow zero across the boundary.
For a plane boundary the image is the optical reflection of the vortex. For example, a pair of equal opposite vortices, moving on a line parallel to a plane boundary, will have a corresponding pair of images, forming a rectangle of vortices, and the path of a vortex will be the Cotes’ spiral
r sin 2θ = 2a, or x−2 + y−2 = a−2;
(10)
this is therefore the path of a single vortex in a right-angled corner; and generally, if the angle of the corner is π/n, the path is the Cotes’ spiral