191-192] EFFECT OF GRAVITATION OF WATER. 319 



7 denoting the gravitation-constant, whence 



Substituting in (10) we find 



where v n is now used to denote the actual speed of the oscillation, 

 and a^ the speed calculated on the former hypothesis of no 

 mutual attraction. Hence the corrected speed is given by 



For an ellipsoidal oscillation (n = 2), and for pjp Q *18 (as in the 

 case of the Earth), we find from (14) that the effect of the mutual 

 attraction is to lower the frequency in the ratio of '94 to 1. 



The slowest oscillation would correspond to n = l, but, as 

 already indicated, it would be necessary, in this mode, to imagine 

 a constraint applied to the globe to keep it at rest. This being 

 premised, it appears from (15) that if p>/? the value of oy 2 is 

 negative. The circular function of t is then replaced by real 

 exponentials ; this shews that the configuration in which the 

 surface of the sea is a sphere concentric with the globe is one 

 of unstable equilibrium. Since the introduction of a constraint 

 tends in the direction of stability, we infer that when p> p the 

 equilibrium is a fortiori unstable when the globe is free. In 

 the extreme case when the globe itself is supposed to have no 

 gravitative power at all, it is obvious that the water, if disturbed, 

 would tend ultimately, under the influence of dissipative forces, to 

 collect itself into a spherical mass, the nucleus being expelled. 



It is obvious from Art. 165, or it may easily be verified inde- 

 pendently, that the forced vibrations due to a given periodic 

 disturbing force, when the gravitation of the water is taken into 

 account, will be given by the formula (10), provided fl n now 

 denote the potential of the extraneous forces only, and cr n have 

 the value given by (15). 



* This result was given by Laplace, Mecanique Celeste, Livre ler, Art. 1 (1799). 

 The free and the forced oscillations of the type n = 2 had been previously investi- 

 gated in his " Becherches sur quelques points du systeme du monde," Mem. de 

 I'Acad. ray. des Sciences, 1775 [1778]; Oeuvres Completes, t. ix., pp. 109,.... 



