1894,] Prof. M. J. M. Hill. On a Spherical Vortex. 219 



IV. " On a Spherical Vortex." By M. J. M. HILL, M.A., D.Sc., 

 Professor of Mathematics at University College, London. 

 Communicated by Professor HENRICI. Received January 

 19, 1894. 



(Abstract.) 



1. In a paper published by the author in the ' Philosophical Trans- 

 actions ' for 1884, on the " Motion of Fluid," part of which is moving 

 rotationally and part irrotationally, a certain case of motion symme- 

 trical with regard to an axis was noticed (see pp. 403 405). 



Taking the axis of symmetry as axis of z, and the distance of any 

 point from it as r, and allowing for a difference of notation, it was 

 shown that the surfaces 



+(!L)!_ 1 = constant, 



where a, c are fixed constants and Z any arbitrary function of the 

 time, always contain the same particles of fluid in a possible case of 

 motion. 



The surfaces are of invariable form. If the constant be negative, 

 they are ring-shaped ; if the constant be zero, the single surface re- 

 presented breaks up into an evanescent cylinder and an ellipsoid of 

 revolution ; if the constant be positive, the surfaces have the axis of 

 revolution for an asymptote. 



The velocity perpendicular to the axis of symmetry is 



the velocity parallel to the axis of symmetry is 



where A; is a fixed constant and Z = dZjdt. 



These expressions (which make the velocity infinitely great at in- 

 finity) 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 of them, provided a rigid, frictionless boundary, 

 having the shape of the limiting surface, be supplied and the boundary 



