PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
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Hence, at the surface of the sphere (XXIL), putting 
r ■=■ a sin d 
z — 7i — a cos 6 
T = 2h sin 6 cos 6 
in = Z — 2k sin® 0 
+ V = 2F cos® 6 ^ k^ — a cos O 'Z -\- — 
P P 
Art. 8, The Irrotational Motion outside the Sphere. 
The velocity potential of a sphere of radius a, moving with velocity 
the axis of :s, at external points, is 
(f) = — cdZ [z — Z) /(2E,^) = — a^Z cos d/(2R®) . . 
where 
112 = ,.3 + {z- Zf 
(see Basset’s ‘Hydrodynamics,’ vol. I., Art. 143). 
Whence 
^ = Sa^Zr {z ~ Z)/(2R5). 
^ = a^Z (3 (2 - Z)3 - B®} /(2R5). 
+ V = [B® {(2 - Z) Z - Z®} + 3 (2 - Z)® Z®]/(2B^) 
P 
- a^Z® [B® + 3 (2 - Z)®]/(8B«) + T . . . . 
where T is an arbitrary function of t. 
Hence, at a point on the surface of the sphere (XXII.), 
^ = f Z sin d cos 0 . 
dr ® 
= Z(1 - |sin®d). 
(XXVII.), 
(XXVIII.), 
(XXIX.), 
(XXX.), 
parallel to 
(XXXI.), 
(XXXII.), 
(XXXIIL), 
(XXXIV.), 
(XXXV.), 
(XXXVI.), 
