678 * 
Proceedings of the Ptoyal Society 
where p denotes one and a half times the volume of the globe, 
and w denotes the kinetic energy of what we may call the internal 
motion of the liquid occupying for an instant in the undisturbed 
motion the space of the rigid globe in the real system. To define 
w, remark that the harmonic analysis proves the velocity of the 
centre of inertia of an irrotationally moving liquid globe to be 
equal to q, the velocity of the liquid at its centre ;* and con¬ 
sider the velocity of any part of the liquid sphere, relatively to a 
rigid body moving with the velocity q. The kinetic energy of 
this relative motion is what is denoted by w. Remark also that if, 
by mutual forces between its parts, the liquid globe were suddenly 
rigidified, the velocity of the whole would be equal to q; and 
that \m(p is the work given up by the rigidified globe and sur¬ 
rounding liquid when the globe is suddenly brought to rest, being the 
same as the work required to start the globe with velocity q from 
rest in a motionless liquid. 
Let P + be the velocity potential at (:e, y, z) in the actual motion of the 
liquid when the rigid globe is fixed. Let a be the radius of the globe, r 
distance of ( x , y, z) from its centre, and ffda- integration over its surface. 
At any point of the surface of the instantaneous liquid globe, the component 
velocity perpendicular to the spherical surface in the undisturbed motion is 
; and hence the impulsive pressure on the spherical surface re¬ 
quired to change the velocity potential of the external liquid from P to P-f-^, 
being — ■£, undoes an amount of work equal to 
in reducing the normal component from that value to zero. On the other 
hand, the internal velocity-potential is reduced from P to zero, and the work 
undone in this process is 
* This follows immediately from the proposition (Thomson and Tait’s 
“ Natural Philosophy,” § 496) that any function V, satisfying Laplace's 
dx 2 dy 2 dz 2 
throughout a spherical space has for its mean 
dP 
value through this space its value at the centre. For — 
dx 
satisfies Laplace’s 
equation. 
