PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
237 
Art. 15. Consideration of the case where the rotationally moving fluid is limited 
Toy the elliysoid^ of revolution 
4 + ^A,. 
a- c“ 
In this case 
Ic 
r = 2^r(z-Z) 
Also 
Jh 
P 
» = z - f {2r - «»') - f (^ - zf. 
+ ^ = i -f)-'Z(z-Z)- f (. - Z)* + f (. - Z)'^ 
+ an arbitrary function of t. 
Now the velocity potential clue to the motion of the ellipsoid, 
moving with velocity Z parallel to the axis of z, is 
= /X (z - Z) j ^ 
(ho 
+ (62 + (c2 -f 
where 
Z = 
(ho 
2^ 
. 0 (C62 + (62 + ^i)l/3 (c3 + ’ 
and e is the parameter of the confocal ellipsoid through the point x, y, z. 8ee 
Basset’s ‘ Hydrodynamics,’ vol. I., Art. 147. 
Then if q be the perpendicular from the centre of the ellipsoid on to a tangent 
plane, the velocity components at the surface are— 
d(f> 
2fjo {z - 
2), 
q-x 
dx 
abd 
rt.2 
B<f) 
1 
<M 
1 
, ftJ 
abd 
62 
d(f) 
2/x(z- 
Z) 
C(z 
Sz “ " cobd' ' c2 + ^ J 0 (fd A (62 + 10f-2 (c3 + nfr2 ’ 
The normal velocity at the surface is therefore 
