

34tt Sir William Thomson on the Motion of 



Hence, if N denote the component velocity of the liquid at (a,b, c) 

 in any direction \, /jl, v, we have 



where 



Let now («, Z>, c) be any point of the barrier surface O (§ 2), and 

 X, fjb, v the direction-cosines of the normal. By (2) of § 2 we 

 see that the part of % due to the motion of the globe is J) Nd<r, 

 or, by (26), 



(4^| + 4)f^^^^^^ ' (28) 

 Hence, putting 



JjF(<27, y, sr, a, b, c)d<r=TJ, 



we see by (8) of § 4 that 



dV Q dXJ dV , om 



," = -Tx> ^"V 7== ~^* ' ' (29) 



Hence with the notation of § 7 (18) for cc, y, , . . instead of 

 ^, 0, . . . ■ 



{y,*}=0, foa?} = 0, {#,y} = 0. 



By this and (25) the equations of motion (22) with (24) become 

 simply 



m ^ =x+ ^-' m J-= Y+ ip m np =z+ -te- < 30 > 



These equations express that the globe moves as a material par- 

 ticle of mass m, with the forces (X, Y, Z) expressly applied to it, 

 would move in a " field of force/' having W for potential. 



12. The value of W is, of course, easily found by aid of sphe- 

 rical harmonics from the velocity-potential, P, of the polycyclic 

 motion which would exist were the globe removed, and which we 

 must suppose known ; and in working it out (see next paragraph) 

 it is readily seen that if, for the hypothetical undisturbed motion 

 q denote the fluid velocity at the point really occupied by the 

 centre of the rigid globe, we have 



W=4/^+^ (31) 



where fi denotes once and a fe half the volume of the globe, 

 and w denotes the kinetic energy of what we may call the in- 

 ternal motion of the liquid occupying for an instant in the un- 



