PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
215 
All the particulars of the motion are placed together in the Table below, in which 
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 
% be then the molecular rotation is given by 
0 T dio 
2ci) ““ • 
dz or 
Also p is the pressure, p the density, and V the potential of the impressed forces. 
The minimum value of p/p + V is IT/p, where U/p must be determined from the 
initial conditions. 
Findher R, 6 are such that 
r = R sin d, 
2 ; — Z = R cos 9. 
The whole motion depends on the following constants : — 
(1.) The radius of the sphere, a. 
(2.) The uniform velocity with which the vortex sphere moves along 
its axis, Z. 
(3.) The minimum value of p/p + V, viz., H/p. 
Rotational motion inside sphere. 
VeHty parallel to axis o£ r | 3Zr (z — Z)l(2a-) . 
' Vel ity parallel to axis of z I Z{5a^ - S (z - Zp - 6?-2}/(2a2) . . . 
At the surface of 
the sphere. 
\Z sin 6 cos 0 . 
Z (I — -f sin2 O') . 
plp\Y-nif>. . 
9Z2 
8a^ [(r2 - - {(z _ Z)2 - + a^] fZ^ cos^ 0 + 
Cabnt function . . . . | 3Zr2{R2 - |a3}/(4a2). 
3Z?-2{R2 — a,2j/(4a2) = constant. 
Sui ces containing the 
s le particles of fluid 
t oughout the motion 
Veljity potential .... j ... . 
Mo/nlar rotation. . . . | 1.5Zr/(4a2) 
J i 
Cji c constant of vortex .! 5aZ . . 
Ki] 
uc energy.j 2ZTrpaPZ^I2\ 
Irrotational motion outside sphere. 
Za^Zr {z - Z)/(2R5) 
fl’5Z{3 {z - Z)2 - R2}/(2R5) 
1Z3 
I + {5 - 4 (a/R)3 - (a/R)«} 1 
_+ 3 cos2 (?{4 (a/R)3 - (a/R)^} J 
- rt3Zr3/(2R3) 
Zr2 (R^ — a^)/(2R®) = constant 
- a^Z {z - Z)I(2W) 
7r/>a3Z2/3 
