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PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
These expressions (which make the velocity infinitely great at infinity) cannot 
apply to a possible case of fluid motion extending to infinity. Hence the fluid moving 
in the above manner must be limited by a surface of finite dimensions. This limiting 
surface must always contain the same particles of fluid. 
Where, as in the present case, the surfaces containing the same particles of fluid 
are of invariable form, it is possible to imagine the fluid limited by any one of them, 
provided a rigid frictionless boundary having the shape of the limiting surface be 
supplied, and the boundary he supposed to move parallel to the axis of 2 with velocity 
Z. Then the above expressions give the velocity components of a possible rotational 
motion inside the boundary. So much was pointed out in the paper cited above. 
2. But a case of much greater interest is obtained when it is possible to limit the 
fluid moving in the above manner by one of the surfaces containing always the same 
particles of fluid, and to discover either an irrotational or rotational motion filling all 
space external to the limiting surface which is continuous with the motion inside 
it as regards velocity normal to the limiting surface and pressure. 
3. It is the object of this paper to discuss such a case, the motion found external to 
the limiting surface being an irrotational motion, and the tangential velocity at the 
limiting surface, as well as the normal velocity, and the pressure being continuous. 
The particular surface (containing the same particles) which is selected is obtained by 
supposing that the constant vanishes, and also that c = a. Then this surface breaks 
up into the evanescent cylinder 
= 0 , 
and the sphere 
+ (, _ Zf = a\ 
The molecular rotation is given by w = bJcrjar, so that the molecular rotation along 
the axis vanishes, and therefore the vortex sphere still possesses to some extent the 
character of a vortex ring. 
The irrotational motion outside a sphere moving in a straight line is known, and it 
is shown in this paper that it will be continuous with the rotational motion inside the 
sphere provided a certain relation be satisfied. 
This relation may be expressed thus ;— 
The cyclic constant of the spherical vortex is five times the product of the radius of 
the sphere and the uniform velocity with ivhich the vortex sphere moves along its axis. 
The analytic expression of the same relation is 
4^- = 3Z. 
This makes 
ixi = I5Zr/(4a“). 
