PROFESSOR W. M. HICKS ON VORTEX MOTION. 
47 
Denote the cyclic constants of the inner and outer portions by /Xj, p,- 
we see that they are respectively 
As before, 
TT 
X vol. 
That is 
Pi = - - f b k, 
IT 
TT ' ' 
Substituting for h, h' in terms of pi, p^ 
V — 1 J _L 
5 1 -T P2 
The result is that a double aggregate is possible. If, however, the size is given 
the ratio of the vorticities must have a special value, and vice versd. In terms 
of the radii it may be shown that 
_ 4 {k - //) W {a - h) {2cd + 4 
45a'* (a + 6) 
Three cases specially invite attention, (l) equal volumes, (2) both parts made of 
similar matter, i.e., vorticities equal, and (3) equal cyclic constants. 
Case i.—Here cd — 26b 
h' 2¥ 1 
A “ “ 6?r'* {(d - h-) - 26' ~ “ 3 X 2'^'** - 4 ’ 
^ ~ -76220 = — f nearly. 
k' 
Case ii.— h' = — h. 
3a® {d~ — Ir) — 46" (a® — 6®) = 0. 
Put a/6 = X, we get 
3x* + 3a3® — 405® — 4 o5 — 4 = 0. 
This has three negative roots ; the positive one is 
€b 
05 = 1-3283 or nearly. 
Case hi.—p = — p' or 
h ¥ k - ¥ 
a? - 6 * — ~ IF 
whence 
_ ceV - 6® (a® - 6®) + la® {b - 6® = 0, 
2a® + 6® — 3a®6 = (^a — h) (2a® — ah — 6®) = (a — 6)® (2a + 6) = 0. 
Equal circulations are therefore impossible. 
