220 Prof. M. J. M. Hill. [Mar. 1, 



be supposed to move parallel to the axis of z with velocity Z. Then 

 the above expressions give the velocity components of a possible 

 rotational motion inside the boundary. So much was painted out in 

 the paper cited above. 



2. Bnt a case of much greater interest is obtained when it is possi- 

 ble 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 mot ion, 

 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 j 

 selected is obtained by supposing that the constant vanishes and 

 also that c = a. Then this surface breaks up into the evanescent ] 

 cylinder 



7^ = 0, 



and the sphere r*+ (z Z)' = o a . 



The molecular rotation is given by ta = 5krja*, so that the mole- 

 cular rotation along the axis vanishes, and therefore the vortex sphere 

 still possesses in a small degree 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 rela- 

 tion 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 which the vortex 

 sphere moves along its axis, 



The analytic expression of the same relation is 



4& = 3Z. 

 This makes ta = l5Zr/(4a a ). 



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

 on p. 221, in which the notation employed is as follows : 



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

 parallel to the axis of z be w, then the molecular rotation is given by 



dt dw 



2ta ~ 3 j- * 

 dz dr 



