PROFESSOR W. M. HICKS OR VORTEX MOTION. 43 
Section ii .—Aggregates with no Secondary Spin {f= 0 ) and with 
Uniform Vorticity. 
6. We begin with the spherical agg’regate, the simplest type of which is the Hill’s 
vortex. The equation for i/; is that given by equation 7, in which oj is put kp where 
k is uniform and y = It is 
I cl'y\r cot 6 dyr 
f f de 
= — Ankp^ = — iTrkA siir d, 
in which 9 is measured from the pole to the equator. A particular solution of this 
is — hirUA sin^ 6. 
In^ 
cZ-’V 1 d-y\r cot 6 d-\Jr _ 
dd ~ 
put i// = Z„ being a function of 6 only. Then 
- cot e‘^ + n(n-l) Z,„ = 0 . 
dd- 
The integral of this is 
Z,.= -smC^ 
where P„_i is a zonal harmonic of degree n — 1. 
Hence the general solution of the equation in xJj is 
B 
Since 
P., = 
xjj = — ^Trkr^ siiV 0 S ( A,.r"' + Z,,^. 
1.3.5 .. . {2n - I) [■ a “ 1) „-o /, . 1 
^r°" ^- 2(2,.- • 1) 
n : 
the values of Z,^ are easily found, excejit for Z, or Zq. It is easily found from the 
direct equation in this case that Zi = Zq = cos 6. The following results are easily 
deduced :— 
Zo =■ siiV 0, Z 3 = 3 sin" 6 cos 6, 
Z, = f (4 sin- 9-5 sin^ 9), sin^ 9 = ^Z.- f,- Z,. 
Consider now first the case of a homogeneous spherical aggregate. In this case 
g 
the functions Z„ apply only to the space outside, and Ar^Z^^ to the space inside. 
Let xj/i denote the value of xp inside and xf/o outside. Hence 
e 2 
