PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
245 
The extreme limits of correspondingf to surfaces inside the vortex sphere are 
and 0, and as # diminishes from to 0, X increases from 0 to 1. 
Putting* 
F (X) = (2 - X)'/’“ f"(l - X siiP defy, 
J 0 
F (X) = - 1 (2 - X)-’/“ f"cos 2(/. (1 - X sin d<f> 
J 0 
= ^ (2 — X)“''^ [ cos 2(f) [(1 — X cos^ — (1 — X siiF d(f). 
Since 0 < (f) < every element of the integral is positive. 
Hence F' (X) is positive ; and, therefore, as X increases from 0 to 1, F (X) increases 
from 77 to CO . 
Hence as cP diminishes from to 0, the time of revolution increases from 
ia-nl^Ti to 00 . 
The fact, that when = 0, the time is infinitely great, may be verified by findipg 
the time along the axis of the vortex sphere from end to end, and the time along a 
meridian from one end of the axis to the other. 
These are 
2a^ r+^‘ d(z-Z) 
2Z )_, (d-{z-Zf’ 
and 
4rt 
‘^TT 
cosec 9 dd, 
0 
both of which are infinitely great. 
This result does not constitute a difficulty, for if a particle anjwvhere on tlie axis of 
the sphere could reach the extremity then it would not be clear along which meridian 
of the sphere it should subsequently move. 
If again the particles on any meridian of the sphere could reach the extremity of 
the axis, there would at that extremity be a collision of the particles coming in from 
all possible meridians. 
