of Edinburgh, Session 1885-86. 
373 
begin to move or it may not. If it does not begin to move of itself, 
give it a very slight motion of rotation round any axis. Generally 
it will begin to move of itself, but it will not do so if the interior 
fluid motion fulfils a definite condition of kinetic equilibrium, and 
therefore if you do not see the containing case beginning to move of 
itself you must set it in motion. When you see it in motion, act 
upon it with a couple in any direction to do some positive work 
upon it, and then suddenly stop it. Left to itself now, it will 
certainly begin to move of itself. When you see it moving again, 
again do some work upon it gradually, and stop its motion suddenly. 
Go on incessantly acting according to this rule. The positive work 
done gradually will exceed the work undone suddenly each time, or 
at all events on the aggregate of a large number of times of repetition 
of the operation. Thus on the whole you will increase the energy 
of the fluid motion without continually giving kinetic energy to the 
containing vessel, as might be the case if you continued always to 
apply a couple in such a direction as to do positive work. Thus 
by going on long enough operating in the manner described we can 
present the containing vessel at rest with the liquid moving inside 
it with any amount of kinetic energy we please. 
A simpler rule suffices for diminishing the internal energy. 
Simply place the containing vessel on flexible imperfectly elastic 
supports, and leave it to itself, or leave it to itself immersed in a 
viscous fluid. Watch it for a while till you see it moving ; or if 
you do not see it beginning to move of itself give it a slight motion, 
then leave it entirely to itself. It will never come to rest unless 
for an instant, and the internal energy will diminish asymptotically 
towards zero. 
I now proceed to prove the propositions regarding fluid motion 
in an ellipsoidal hollow referred to above. 
I. Irrotational motion of liquid in a rigid ellipsoidal shell. 
Given the motion of the boundary : required the motion of the 
contained liquid. 
Let <sr, p, o-, be the component velocities of the shell, and let cf> 
be the velocity potential of the corresponding determinate * motion 
* Wm. Thomson, “On the Yisviva of a Liquid in Motion,” Camb. and Dub. 
Math. Journal, 1849 ; or Thomson and Tait’s Natural Philosophy, secs. 312 
and 317, example (3). 
