ANALYSIS OF DISTURBING FORCES. 367 



The ' constant ' C is to be determined by the consideration that, on account 

 of the invariability of volume, we must have 



SSCdS = .................................... (xi), 



where the integration extends over the surface of the ocean. If the ocean 

 cover the whole earth we have (7=0, by the general property of spherical 

 surface-harmonics quoted in Art. 88. It appears from (vii) that the greatest 

 elevation above the undisturbed level is then at the points ^ = 0, ^ = 180, i.e. 

 at the points where the disturbing body is in the zenith or nadir, and the 

 amount of this elevation is \H. The greatest depression is at places where 

 ^ = 90, i.e. the disturbing body is on the horizon, and is \H. The greatest 

 possible range is therefore equal to H. 



In the case of a limited ocean, C does not vanish, but has at each instant a 

 definite value depending on the position of the disturbing body relative to the 

 earth. This value may be easily written down from equations (x) and (xi) ; 

 it is a sum of spherical harmonic functions of A, a, of the second order, with 

 constant coefficients in the form of surface-integrals whose values depend on 

 the distribution of land and water over the globe. The changes in the value 

 of (7, due to relative motion of the disturbing body, give a general rise and 

 fall of the free surface, with (in the case of the moon) fortnightly, diurnal, and 

 semi-diurnal periods. This * correction to the equilibrium-theory,' as usually 

 presented, was first fully investigated by Thomson and Tait*. The necessity 

 for a correction of the kind, in the case of a limited sea, had however been 

 recognized by D. Bernoulli f. 



d. We have up to this point neglected the mutual attraction of the par- 

 ticles of the water. To take this into account, we must add to the disturbing 

 potential Q the gravitation-potential of the elevated water. In the case of an 

 ocean covering the earth, the correction can be easily applied, as in Art. 192. 

 Putting 7i = 2 in the formulae of that Art., the addition to the value of fi is 

 -f p/Po- # ; and we thence find without difficulty 



It appears that all the tides are increased, in the ratio (1 fp/po)" 1 - If we 

 assume p/p = '18, this ratio is T12. 



e. So much for the equilibrium-theory. For the purposes of the kinetic 

 theory of Arts. 206 214, it is necessary to suppose the value (x) of to be 

 expanded in a series of simple-harmonic functions of the time. The actual 



* Natural Philosophy, Art. 808 ; see also Prof. G. H. Darwin, " On the Cor- 

 rection to the Equilibrium Theory of the Tides for the Continents," Proc. Roy. Soc., 

 April 1, 1886. It appears as the result of a numerical calculation by Prof. H. H. 

 Turner, appended to this paper, that with the actual distribution of land and water 

 the correction is of little importance. 



t Traite sur le Flux et Reflux de la Her, c. xi. (1740). This essay, as well as the 

 one by Maclaurin cited on p. 322, and another on the same subject by Euler, is 

 reprinted in Le Seur and Jacquier's edition of Newton's Principia. 



