224 
PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
V 
+ V = 1 a COS ^ Z - I Z3 + f cos'- ^ Z^ + T . . (XXXYIL). 
The value of the current function if/, corresponding to the velocity potential of 
(XXXI.) is 
i// = - a3Zr2/(2R3).(XXXYIIL). 
If X = const, be a family of surfaces containing the same particles of fluid 
0A, I cyjr cX 
dt r dz dr 
1 0-\/r Sx, 
T dr dz 
= 0 . 
. (XXXIX.). 
An integral of this equation is 
for Z being constant. 
therefore 
k = ifj ^ Z 
^ ^ ^ 7\ 
dt~ dt ~ dz^ ’’ 
dx d'\lr ry 
A- + 
dx _ d-^Jr 
0^’ 
0X 1 dyjr dx 1 d-^lr dX 
dt r dz dr r dr dz 
(XL.), 
1 v^jr dyfr 
r dr 0s 
= 0. 
Hence the surfaces X = const, are 
r- 
z(l - = const. (XLL). 
Art. 9. The Conditions for the continuity of the rotational and irrotational motions. 
In order that the motion inside the sphere (XXII.) may be continuous with that 
outside, the equations (XXYIII.) and (XXXY.) must make r = d^jdr. 
Therefore 
2Z: = I Z.(XLIL). 
The equations (XXIX.) and (XXXYL) must make w = 0^/02. 
