■PHOPESSOR W. M. HICKS OH VORTEX MOTIOH. 
53 
We work out the case for a dyad a^^gregate, in which ko = — ^i, 
A{r) = 27T{{^a\-^a^ky--ik,r^}, 
f, (r) = 277 [ - i k^ik^ + k,a\/r + i k,r^}, 
and 
E = ^ {iV ( 2 «i — «i) a\ — ^ fd + iT 0^2 (ai - «?) 
-h (cfr2 — «?) — 5-7 («2 — «0 
— yg ^ ( 3 '^<'1 t ^2®i H" I'-y OS'’] 
o3'/( A/l dm O^ll 7\ 
= "4h“ ~ ^2^1 + 7 «2) 
^ _ ?>2irk\ . 
= 1‘945 X 7 ;—^ a\. 
4o X 7 
If the two parts had been single monads their combined energy (when far apart) 
would have been 
E 
, ro. 327r/d , 
^ 45x7^'^- 
The energy when combined is therefore greater than when they are separate. 
11. It may not be out of place to make a short digression here as to the relation 
of a Hill’s vortex to the vortex rings which have been investigated in previous parts 
of these researches. As is known the translation velocity of an ordinary ring 
decreases as the energy increases, and formulm are given in a former paper'" whereby 
those quantities can be calculated for comparatively thick rings up to R/r = 4 with 
considerable accuracy, and possibly further. Here R is the radius of the equatorial 
axis and r the mean radius of the section of the ring. Refer all measurements to the 
spherical form, and let c denote its radius, Vo its velocity of translation, and Eq its 
energy. Take now a ring of the same volume and circulation as the sphere, and let 
V and E denote its translation velocity and energy. We get the following value of 
E/Eo, V/Vo for different apertures. 
R 
r 
R 
c 
r 
c 
V 
w 
E 
Eo‘ 
100 
•199 
176 
50 
S'OO 
-162 
•282 
95 
10 
2-77 
-277 
•593 
20-8 
5 
1-745 
•349 
•784 
10-25 
4 
1-500 
-375 
•856 
8 
3 
1-239 
•413 
■946 
6 
* “ Researches in the Theory of Vortex Rings,” Part II., p. 757, ‘ Phil. Trans.,’ 1885, Pai't II. 
