190 
MR. W. M. HICKS ON THE STEADY MOTION AND 
12. On account of the constant surface velocity along a hollow vortex, with fluid 
streaming past it, it is very easy to determine the energy of the motion, when the 
ring moves through a fluid otherwise at rest. For § 8 the energy is given by 
where ds is an element of the arc of the cross-section, l its length, and U' the velocity 
along the surface regarded as for the moment fixed in space. It is therefore the 
component of Y along the surface and U, that is 
u^u-f y 1 "— 
r 
Also 
t//=ijrVp 2 + const 
=¥^(p 2 —' Ka2 )+ x l i o 
Hence, to the first order, where the section is a circle, the energy is 
JR -o\i. V. 
£'(u+V^)(4-h« s +iVp' : 
p=R+r cos 6 
3 \rd6 
where 
Therefore the energy is 
nrC* [U{4-f iY(R 3 -Xa 3 )} + (iUYr 3 +V 2 Rr) cos 2 0]d6 
Jo 
= 2 7 rV[U{^ 0 +iY(R 2 -Xa 2 )}+iYr(iUr+YR)] 
But 
U: 
47 rak 
v =fb< L -i) 
Substituting these values, the energy is 
Now to the order of approximation of circular section X=l, r—2ak, R=«, and the 
energy is ^/u. 2 a(L — 2), which is the same as for a rigid tore at rest. If the shape be 
