Michell has discussed also the hollow vortex stationary inside a polygon (Phil. Trans., 1890); the solution is given by
ch nΩ = sn w, sh nΩ = i cn w
(11)
so that, round the boundary of the polygon, ψ = K′, sin nθ = 0; and on the surface of the vortex ψ = 0, q = Q, and
cos nθ = sn φ, nθ = ½π − am s/c,
(12)
the intrinsic equation of the curve.
This is a closed Sumner line for n = 1, when the boundary consists of two parallel walls; and n = ½ gives an Elastica.
44. The Motion of a Solid through a Liquid.—An important problem in the motion of a liquid is the determination of the state of velocity set up by the passage of a solid through it; and thence of the pressure and reaction of the liquid on the surface of the solid, by which its motion is influenced when it is free.
Beginning with a single body in liquid extending to infinity, and denoting by U, V, W, P, Q, R the components of linear and angular velocity with respect to axes fixed in the body, the velocity function takes the form