Michell has discussed also the hollow vortex stationary inside a polygon (_Phil. Trans._, 1890); the solution is given by ch n[Omega] = sn w, sh n[Omega] = i cn w (11) so that, round the boundary of the polygon, [psi] = K´, sin n[theta] = 0; and on the surface of the vortex [psi] = 0, q = Q, and cos n[theta] = sn [phi], n[theta] = ½[pi] - 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 [phi] = U_[phi]1 + V_[phi]2 + W_[phi]3 + P_[chi]1 + Q_[chi]2 + R_[chi]3, (1) where the [phi]'s and [chi]'s are functions of x, y, z, depending on the shape of the body; interpreted dynamically, C - [rho][phi] represents the impulsive pressure required to stop the motion, or C + [rho][phi] to start it again from rest. The terms of [phi] may be determined one at a time, and this problem is purely kinematical; thus to determine [phi]1, the component U alone is taken to exist, and then l, m, n, denoting the direction cosines of the normal of the surface drawn into the exterior liquid, the function [phi]1 must be determined to satisfy the conditions (i.) [nabla]²[phi]1 = 0. throughout the liquid; (ii.) d[ph]1/d[upsilon] = -l, the gradient of [phi] down the normal at the surface of the moving solid; (iii.) d[ph]1/d[upsilon] = 0, over a fixed boundary, or at infinity; similarly for [phi]2 and [phi]3. To determine [chi]1 the angular velocity P alone is introduced, and the conditions to be satisfied are (i.) [nabla]²[chi]1 = 0, throughout the liquid; (ii.) d[chi]1/d[upsilon] = mz - ny, at the surface of the moving body, but zero over a fixed surface, and at infinity; the same for [chi]2 and [chi]3. For a cavity filled with liquid in the interior of the body, since the liquid inside moves bodily for a motion of translation only, [phi]1 = -x, [phi]2 = -y, [phi]3 = -z; (2) but a rotation will stir up the liquid in the cavity, so that the [chi]'s depend on the shape of the surface. The ellipsoid was the shape first worked out, by George Green, in his _Research on the Vibration of a Pendulum in a Fluid Medium_ (1833); the extension to any other surface will form an important step in this subject. A system of confocal ellipsoids is taken