300 CELESTIAL MECHANICS. I. vH. 35. 



and this, divided by d^^^ becomes equivalent to] the ex- 

 pression contained in the proposition. 



373. Corollary. At the surface of the 

 sea, we have r^^Q [-r^ — 2n sin cos ^ tt) + ^'^^'^ 



(sin«5 TT^ H- 2n sin cos o ^ + 2nsiim-T7)=^ — 



g^y'+W: g being the force of gravity, a^y' the 

 elevation above the surface of equihbrium, 

 and aSP the part of g F which relates to the 

 disturbing forces only. 



At the external surface of the fluid, we have ^p = 0, 

 and in the state of equilibrium 



0=^n^d i(r-has) sin {9 + au)l» + (SF), (^V) being 



the value of S'F which belongs to this state: [since the cen- 

 trifugal force, together with the force contained in F, must 

 in this state balance each other; and the quantities s, u, 

 and V being constant, the first member of the equation (L) 

 must necessarily vanish.] If the fluid in question be the 

 sea, the variation {^V) at its surface will be the force of 

 gravity multiplied by the element of its direction: and 

 calling this force g, and making ai/ the elevation of a par- 

 ticle of the surface above the surface of equilibrium, which 

 may in this case be considered as the true level of the sea ; 

 it will be evident that the variation (d'V) will be increased, 

 in the state of motion, by the quantity — ccg^i/', because the 

 force of gravity acts very nearly in the direction of y\ and 

 tends towards its origin [ : the y' here intended being 

 however very different from the y of the former part of the 

 proposition, which is an immoveable line, and the force 



