120 Proceedings of Royal Society of Edinburgh. [sess, 
particular case, a figure of revolution with its axis vertical for the 
containing vessel and let the given motion he rotation round this 
axis suddenly commenced and afterwards maintained with uniform 
angular velocity. The initial kinetic energy will be zero for each 
of the three substances. The inviscid liquid will remain for ever 
at rest ; the water will acquire motion according to the Fourier 
law of diffusion of which we know something for this case by 
observation of the result of giving an approximately uniform 
angular motion round the vertical axis to a cup of tea initially at 
rest. The jelly will acquire laminar wave motion proceeding- 
inwards from the boundary. But in the present communication 
we confine our attention to the case of inviscid liquid. 
The now well-known solution* of the minimum problem thus 
presented, when the bounding surface is simply continuous, is, 
simply : that the initial motion of the liquid is irrotational. 
That the initial motion must be irrotational f is indeed obvious, 
when we consider that the impulsive pressure by which any 
portion of the liquid is set in motion is everywhere perpen- 
dicular to the interface between it and the contiguous matter 
around it, and therefore the initial moment of momentum round 
any diameter of every spherical portion, large or small, is zero. 
But that irrotationality of the motion of every spherical portion 
of the liquid suffices to determine the motion within a simply con- 
tinuous boundary having any stated motion, is not obvious without 
mathematical investigation. 
Whether the boundary is simply continuous, or multiply con- 
tinuous, irrotationality suffices to determine the motion produced, 
as we now suppose it to be produced, from rest by a given motion 
of the boundary. 
Now in a homogeneous liquid acted on by no bodily force, or 
only by such force (gravity, for example) as could not move it 
when its boundary is fixed, the motion started from rest by any 
movement of the boundary remains always irrotational, as we 
knoAV from elementary hydrokinetics. Hence, if at any time the 
boundary is suddenly or gradually brought to rest, the motion of 
* Thomson and Tait’s Natural Philosophy, sec. 312. 
f That is to say, motion such that the moment of momentum of every 
spherical portion, large or small, is zero round every diameter.. 
