198 WAVES IN LIQUIDS. [CHAP. VII. 



solution of the equation of continuity y*&amp;lt; = 0, subject to the 



condition that -3- = when r a, is 

 dr 



(45), 



where S n is the general surface-harmonic of order n. The 2w + 1 

 arbitrary constants which the general expression for S n contains 

 are functions of ,-to be determined. The formula for the pressure 

 is, if we neglect squares and products of small quantities, 



where p is the density x&amp;gt;f the fluid, Fthe potential due to the joint 

 attraction of the earth and the sea. If we assume as the equation 

 of the free surface at time t 



(47), 



where T n is a spherical harmonic of order n, we have*, at this 

 surface, 



r 



Here E denotes the total mass of the earth and sea, viz. 

 E = jiraV + |TT (b 3 - a 3 ) p, 



if cr be the mean density of the earth. Substituting in (46), ex 

 panding, and omitting as before the squares of small quantities we 

 find, at the free surface, 



W 



But at the free surface p = const. If we put g = , this con 

 sideration gives 



3 pb 5 



* Pratt, Figure of the Earth, c. 3. It is assumed of course that E, p, &amp;lt;r are all 

 expressed in astronomical units. 



