242 
PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
Hence the equations 
0T 
, 9t 
+ r ^ w 
0T 
S7 
become 
Therefore 
dio , dw , du) 
~ 7ir{Z — 1) — 2mn7'^ = — 
- 2n (z _ Z) (/ - Z) + 2 n® {z - Zf = - + v| 
7 + V = 4 + 4 Vi (Z - /) + 7i{l - Z){z - Zf - ^ n~ {z - Zf 
+ an arbitrary function of t. 
Th is value of lyjp -J- 7 is not continuous with the value of + V for the motion 
outside the ellipsoid. 
Summary of Kesults. 
A. Rotational Motion inside the Sphere r^ + (2 — Zf = a~. 
Velocity parallel to axis of r = 3Zr (z — Z)/(2cr) 1 ^ 
Velocity parallel to axis of z = Z [5rr — 3 (s — Z)"— 6r“j/(2cr)J 
-7 + V = UZ^ [(r^' - h erf - {{z- Zf - edf + a^]/(8a^) + ^ . (XLVL). 
Current Function i// = 3Zr'{iF — -|(r]/(4cr).(XLVIL). 
Surfaces containing the same particles of Huid 
3Zr~;Il^ — cr]/(4fr) = const.(XLVIll). 
Molecular Rotation = 15Zr/(4a~).(XLIX.). 
Cyclic Constant of Vortex = 5uZ.(L.). 
