313 CELESTIAL MECHANICS. I. vH. 37- 



mosphere becomes /g•hlf=zC^- F+^nV^ sin- 9. Now at 

 the surface of the sea, the value oF V, expressing the 

 force, must be the same for a particle of air as for the par- 

 ticle of water in contact with it, the same forces acting in 

 both cases : but from the condition of the equilibrium of 

 the sea, we have V + -J/i^ r^ sin ^9 constant; consequently 

 /o-hlp must be constant, and f, the density of the stratum 

 of air, contiguous to the sea, must be every where the same 

 in the state of equilibrium. [It is not intended by this 

 constancy of the force, to imply that gravitation is equal 

 throuohout the surface of the sea, but that the pressure on 

 it must be every where equal.] 



If we make R equal to the part of the radius r compre- 

 hended between the centre of the spheroid and the sur- 

 face of the sea, and / the part between that surface and a 

 particle of air elevated above it, we may consider / as the 

 vertical height of the particle above the surface, which it 



will be with only an error of the order I — r'l : R; and 



quantities of this order may be neglected without iuaccu- 



dP ddT' 



racy. Then if F', -; — , and — r-r- be the values of these 

 dr dr^ 



quantities at the surface of the sea, we shall have, for the 



dV r'2 ddF' 

 elevation /, F - P -\- r' ~ + ^ , ■- , [by Taylor s 



dr 2 dr^ 



theorem (247)] and the equation lgh]p=: C-\-V+^ n^r^ smH 



dP r'^ ddP 



will become IqhXo = C + F' + / h -— . ----— + -i- w« 



dr 2 dr^ 



m sin H + w^ -Rr' sin ^ : and for the value of V at the 



surface of the sea, we have F -\- ^n^R^ sin^^zra constant 



quantity : the effect of gravitation at this surface being 



^"TT* ~"'** -R ^^^ *^» which we may call g\ The quantity 



