242-243] OCEAN OF UNIFORM DEPTH. 441 



If p=p , we have ^ = as we should expect. When p>p the value of o^ 

 is imaginary ; the equilibrium configuration in which the external surface 

 of the fluid is concentric with the nucleus is then unstable. (Of. Art. 192.) 



If in (ix) we put 6=0, we reproduce the result of the preceding Art. If, 

 on the other hand, the depth of the ocean be small compared with the radius, 

 we find, putting b = a-h, and neglecting the square of A/a, 



provided n be small compared with a//i. This agrees with Laplace's result, 

 obtained in a more direct manner in Art. 192. 



But if n be comparable with a/k, we have, putting n=ka, 



so that (ix) reduces to <r 2 =#tanhM ................................. (xi), 



as in Art. 217. Moreover, the expression (ii) for the velocity-potential 

 becomes, if we write r=a+z, 



< = <#>! cosh (2 + A) .............................. (xii), 



where <p x is a function of the coordinates in the surface, which may now be 

 treated as plane. Cf. Art. 236. 



The formulae for the kinetic and potential energies, in the general case, are 

 easily found by the same method as in the preceding Art. to be 



and F-taal-JJtfiS ..................... (*iv). 



The latter result shews, again, that the equilibrium configuration is one of 

 minimum potential energy, and therefore thoroughly stable, provided p<p . 



In the case where the depth is relatively small, whilst n is finite, we obtain, 

 putting 6 = a-A, 



whilst the expression for V is of course unaltered. 



If the amplitudes of the harmonics H be regarded as generalized co- 

 ordinates (Art. 165), the formula (xv) shews that for relatively small depths 

 the ' inertia-coefficients ' vary inversely as the depth. We have had frequent 

 illustrations of this principle in our discussions of tidal waves. 



