Solid Bodies through a Liquid. 341 



disturbed motion the space of the rigid globe in the real system. 

 To define w, remark that the harmonic analysis proves the velo- 

 city 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 consider the velocity of any part of the liquid sphere, rela- 

 tively 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 \mq^ is the work given up by the rigidified globe 

 and surrounding 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 + i/r be the velocity-potential at (x,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 fr^o* integration over its surface. At any point of the sur- 

 face 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 required to change the velocity-potential of the external 

 liquid from P to P + ^, being —^r } undergoes an amount of 

 work equal to rr* \ ^p 



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 



d(TV w 



Hence 





w - 2 



The condition that with velocity-potential P-fi/r there is no flow 

 perpendicular to the spherical surface gives 



cf +m„=° « 



* This follows immediately from the proposition (Thomson and Tait's 

 ' Natural Philosophy/ § 496) that any function V, satisfying Laplace's 



equation —t~t "•* ~j~2 ' "jT throughout a spherical space, has for its 



mean value through this space its value at the centre ; for -r~ satisfies 

 Laplace's equation. 



