214-215] STABILITY OF THE OCEAN. 363 



before. Since &quot;^ - ?i 2 w 2 is constant over the undisturbed level 

 (z = 0), its value at a small altitude z may be taken to be cjz + const., 

 where, as in Art. 206, 



Since f/dS = Q, on account of the constancy of volume, we find 

 from (1) that the increment of V T Q is 



iffgraa (3). 



This is essentially positive, and the equilibrium is therefore 

 secularly stable*. 



It is to be noticed that this proof does not involve any 

 restriction as to the depth of the fluid, or as to smallness of the 

 ellipticity, or even as to symmetry of the undisturbed surface with 

 respect to the axis of rotation. 



If we wish to take into account the mutual attraction of the 

 water, the problem can only be solved without difficulty when the 

 undisturbed surface is nearly spherical, and we neglect the varia 

 tion of g. The question (as to secular stability) is then exactly the 

 same as in the case of no rotation. The calculation for this case 

 will find an appropriate place in the next chapter. The result, as 

 we might anticipate from Art. 192, is that the ocean is stable if, 

 and only if, its density be less than the mean density of the Earth*. 



* Cf. Laplace, Mecaniquc Celeste, Livre 4 mo , Arts. 13, 14. 



