304 CELESTIAL MECHANICS. I. vH. S6. 



with regard to the variable quantity r, between the sur- 

 face of the solid spheroid and that of the sea, it is obvi- 

 ous that the value of s will be a function of 9, 'zjt, and t, 

 independent of r, together with a very small function of r, 

 standing in the same relation to u and v, as y does to r, 

 Now at the surface of the solid covered by the sea, when 

 the angles 9 and 'zd- are changed into B + au and nt-\-'m + av, 

 it is easy to see that the distance of a particle of water con- 

 tiguous to that surface, from the centre of gravity of the 

 earth, can only vary by a quantity which is very small with 

 respect to au and aVy and which is of the same order as the 

 products of these quantities into the eccentricity of the 

 spheroid covered by the sea ; consequently the function, 

 independent of r, that enters into the expression of 5, must 

 therefore be of the same order, and very minute, so that 

 we may in general neglect s as inconsiderable in compari- 

 son with u and v, [Thus if the sea were 4 miles deep, y 

 would be about yoVo ^^ ^> ^^^ ^^® ascent and descent of a 

 particle even at the surface of the sea would in general be 

 little more than j^oq- of its horizontal motion, supposing 

 the neighbouring particles, for a considerable extent in 

 comparison with the radius, to be moving in the same di- 

 rection.] We may therefore omit the quantity d^ in the 

 equation of article 373, and it affords the equation of this 

 proposition. 



376. Theorem. The equation of conti- 

 nuity becomes ,=-^^-'J^^-^^, (N) 

 y being the elevation above the surface of 

 equihbrium, and y the depth of the sea. 



The equation (X), article 372, which is applicable to 

 every particle of the fluid, affords us, in the case of equi- 



