PROFESSOR W. M. HICKS OH VORTEX MOTION. 
G7 
or 
E = 
TTfi-a 
9 ' 
X,- 
+ siiP \ 
(S-iX — sin X)- 
( 19 ). 
The energy due to translation is that due to the bodily translation of the sphere 
+ 4 the same. 
The velocity of translation is 
U = 
Hence this part of the energy is 
Therefore total energy is 
iTfjra 
IJ. .T' — i X sin X 
a X (Six — sin X) 
3. i 4 ..^^,3TT2 
2 • 2 • 3 ^ • 
1 ) J'" + i ^ ( V 'T' — I sin X ) 
(Six - sin X)- [ \ ^ X- / ■ ■ ■ \ X 
,, , {2 (i - ^) J'" - I ^ + *- 
(&^X — Sin X)- L J oX •' 
A verification is afforded by putting X = 0 (Hill’s vortex). Then 
(20). 
(S^X — sin X)- = X*^. 
Large bracket = 5 ~t^ 9 T 9 
E = 3% 77/4 Vg 
which is correct. 
The preceding formulae refer to the whole aggregate. When, however, X > the 
lowest k,, there are more than one component, and it will be well to give the 
requisite formulae for each of these separately. Denote Xr„/« by y„, where is the 
radius of the vrth interface from the centre. Also for shortness let 8 (x) denote the 
function Six — sin x. Then 
_ B(y„) - S(y„_d 
fi S (X) 
( 21 ). 
77 
S(y,) - s (//„_,) 
IT _ T'■' 
I’J ,1 X,, 0 j . 
• (22), 
J,„ denoting the value of J at the equatorial axis= 
M„ = 
VI fj. 
X“(SiX — sin X) ^ y„_j 
J_ 
x" S(y„) - S(y„_i) 
j ( y sill !/ — if cosy — y* cly 
^yJ {ya) — 3?/,_iJ (y„_i) - 'iji sin y„ + sin y,_, 
_ yl - yl-i . J (x; (^ 
5 * X- J 
K 2 
