44 
PROPESSOB W. M. HICKS OX VORTEX MOTIOX 
i//i = — ^Trkr^ sin''" 6 + 
Let a denote the radius of the sphere. Along the bonndaiy of the sphere xfji = i//,, 
and also d^ijclr = dxjjoldr. ExjDressing siid^ in terms of Zo and Z^, 
^1= — ^7rJcr^Z2 + Y^irkr^Zi + SA„r“Z„. 
The term in -^^nh'^Zi may be supposed merged in A^r^Zi, and may therefore be 
treated as absent. The conditions 
^1 = ^2 
when r = a 
dr 
dr 
J 
give 
and for m = 2 
Ai = 0 , A,I = 0 when n > 2 , 
Bi = 0 , = 0 when n > 2, 
KoO? — ^Trkcd = 
B., 
a 
2 As « — ^Trkcd = — ^ 
Hence 
and 
As = ^Trkcd, Bs = -^^Trkak, 
xfji = 27Tk [\a“r' — y?’'') siid^ 
\jj2 = "^T^k y siid0. 
The velocity along the normal to the aggregate is 
1 d’Kjr 2 _ 4 7'^ 73 
T- — = -fska- cos 6. 
Iirp rdd ^ ^ 
Hence the aggregate moves forward through the surrounding fluid with a velocity 
V = -^^ka\ 
Beferred to the aggregate at rest therefore 
xp^ — ^irkr- i^cr — r~) siir^. 
t* 
The cyclic constant (p.) is the circulation taken round a meridian section, up the 
polar axis and down outside. It is the sum of the circulation round the elementary 
areas of which the section is composed. Hence ' A. ' 
