PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
235 
When # = cc^, the section of the surface, by a plane through the axis, shrinks 
into a point ellipse whose major axis, which is parallel to the axis of 2 , is double of its 
minor axis. 
As # diminishes from to 0, the surfaces increase in size until finally they 
become merged in the sphere = 0, and the evanescent cylinder — 0. 
Table 11. gives the form of the surfaces 
7- {1 ~ = d:\ 
which are outside the sphere, and which always contain the same particles of fluid 
throughout the motion. 
When d? = 0, the surface merges in the evanescent cylinder = 0, the sphere 
1 — a/R = 0, and the imaginary locus 1 + + («/Tl)^ = 0. 
As d increases from 0 to co, the surfaces tend to become cylinders. It may be 
noticed that the surface [1 — (a/R)^} = d^ has the asymptotic cylinder r = d. 
The greatest distance of this surface from the axis is found by putting 2 — Z = 0, 
and, therefore, R = r. Hence, the greatest distance is a root of the ecpiation 
When r = 10 a is a root of this equation. 
2 n 2 
