307-308] OSCILLATIONS OF A VISCOUS SPHEROID. 565 



mass of liquid about the spherical form. The principal result of 

 the investigation can, however, be obtained more simply by the 

 method of Art. 301. 



It was shewn in Arts. 241, 242, that when viscosity is neglected, the 

 velocity-potential in any fundamental mode is of the form 



*-J~&.oos(rl+f) ........................... (i), 



where S n is a surface harmonic. This gives for twice the kinetic energy 

 included within a sphere of radius r, the expression 



2n+i 



r \2n+i f r 

 -^ JJ 



if So; denote an elementary solid angle, and therefore for the total kinetic 

 energy 



T=^p7iajjS n 2 d^.A 2 cos 2 (a-t+} ..................... (iii). 



The potential energy must therefore be given by the formula 



V=%pnattS n *d&.A 2 am*((rt + e ) ..................... (iv). 



Hence the total energy is 



* ........................... (v). 



Again, the dissipation in a sphere of radius r, calculated on the assumption 

 that the motion is irrotational, is, by Art. 287 (vi), 



Now r>fdw4,dm ..................... (vii), 



each side, when multiplied by p8r being double the kinetic energy of the fluid 

 contained between two spheres of radii r and r + 8r. Hence, from (ii), 



Substituting in (vi), and putting r = a, we have, for the total dissipation, 



2F=2n(n-l)(2n + l) I \S^dw. A 2 cos?(<rt + e) ...... (viii). 



The mean dissipation, per unit time, is therefore 



2F=n(n-l)(2n + I) f fs n 2 d&.A* .................. (ix). 



If the effect of viscosity be represented by a gradual variation of the 

 coefficient J, we must have 



~)=~2F .............................. (x), 



