696 EEPOBT— 1893. 



SATURDAY, SEPTEMBER 16. 



The following Reports and Papers were read : — 



1. Report of the Committee on Mathematical Tahles. — See Reports, p. 227. 



2. Report of the Committee on the Pellian Equation. — See Reports, p. 73. 



3. On a Spherical Vortex. By M. J. M. Hill, M.A., D.Sc, Professor of 



Mathematics at University College, London. 



In a paper by the author published in the ' Philosophical Transactions of the 

 Eoyal Society' in 1884, on the Motion of Fluid, part of which is moving' 

 rotationally and part irrotationally, a certain case of motion symmetrical with 

 regard to an axis was noticed. 



Takiog the axis of symmetry as the axis of z, and the distance of any point 

 from it as r, it was shown that the surfaces 



ar"' {z-Zy- + b (_!•• - ^c^)"- = const., 



where «, b, c are fixed constants, Z any arbitrary function of the time, contain 

 always the same particles of fluid in a possible case of motion. 



If the constant be less than j6c^ these surfaces are rings. 



The autlior has not succeeded in determining an irrotational motion on one 

 side of any of these surfaces continuous with the rotational motion on the other side, 

 except in the particular case in which b = a, and the constant on the right-hand 

 side = ^bcK 



The object of this paper is to discuss this case. 



In it tlie surface containing the same particles of fluid breaks up into the 

 evanescent cylinder 



r= = 0, 

 and the sphere 



7-''+{z-Zy- = c\ 



The molecular rotation is given by Soj = 10«r, so that the molecular rotation 

 along the axis vanishes, and therefore the vortex sphere stiU possesses the character 

 of a vortex ring. 



The irrotational motion outside a spliere 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 sj}herical vortex is Jive times the product of the 7-adius 

 of the sphere and the uniform velocity with which the vortex sphere moves along its 

 axis. 



The analytic expression of the same relation is 



4ac- = 3i. 

 This makes 



2co = 15Z?V(2c^). 



All the particulars of the motion are placed together in the following table. 

 The notation employed is as follows : — 



If the velocity parallel to the axis of r be r, and the velocity parallel to the 

 axis of s be w, then the molecular rotation is given by 



o dr dxo 

 ZQ) = — — — . 

 dz dr 



Also p is the pressure, p the density, and V the potential of the impressed forces. 



