PROFESSOR W. M. HICKS OK VORTEX MOTION. 
45 
fji = % (elementary circulations) = ^2(o dA. = k%2p dK 
Ic k ')uJc 
= - %2iTp dK = - X volume of aggregate = —^ • 
Thus 
TTfl fX 
OJ — Icp = - P \ ; 
' m ' ba 
which are Hill’s results obtained by direct methods. 
7. Heterogeneous Aggregates .—We may, however, superpose on an aggregate such 
as the foregoing other spherical layers of dilferent vorticities. It will be advisable to 
consider fii-st the case where there is one such layer of vorticity determined by 
(say) We may call them dyads. In this outer portion both terms in Ar"' and 
g/^n-i appear. Let i// 2 , denote the stream functions for each part and for 
the surrounding fluid. Then 
xjji = — -f nkAZ.^ + SK,,r'''Z-„_, 
nk'r'Z, + 2 (A.,y- + ^1:;) Z,., 
-^ = 2 ,-!i; Z... 
Let a, h denote the radii of the two spherical surfaces (a > 6), and apply the same 
conditions as before to the two surfaces. 
Again all the co-efEcients vanish except for n = 2, and there results 
t) / "I 
Koh- - I ttMA = + -f - I 
o 6 ^ I 
2kdj - ^TrkV = 2K^h - ^ - ^rrk'h^ 
J 
and 
^^2 A ' I 1^2 o 1 t i ^ 
~ = Ao a 4- — — ink a \ 
a a ^ 1^ 
- 4 = 2A,'a - 4 - I TrkV 
The first two give at once 
the last two 
A 2 — %TTa^k'; 
B.2 = 1^5-77 {{k - k') V + 
A, = f 77 \k‘a^ 4-\k - F) ¥]-. 
also 
