I 



- OF THE MOTIONS OF FLUIDS. 297 



If the external surface of the mass is at liberty, we have 

 here dp— 0, and consequently = 8 F+w-(j/Sy + zSz) ; con- 

 sequently the result of all the forces that acton the external 

 surface must be perpendicular to that surface; it must 

 also be directed towards the interior of the fluid: and 

 when these conditions are fulfilled, a homogeneous fluid 

 mass may be in equilibrium, whatever may be the form of 

 the solid which it covers. 



This case is one of those in which the variation u^x-i- 

 v^j/+wdz is not exact; for this variation becomes equal to 

 --w(zSy— ySz) : and zdy — y^z is not an integrable quan- 

 tity. Consequently in the theory of the tides we cannot 

 suppose the variation 8^ to bo exact, since it is not so in 

 the very simple case of the sea having no other motion 

 than its rotation in common with the earth. 



§ 35. Determination of the very small oscillations of a 

 homogeneous fluid, covering a revolving spheroid. P. 98. 



372. Theorem. If rbe the primitive dis- 

 tance of a particle from the centre, 9 the angle 

 formed by r with the axis x, -ar the angle 

 formed by the plane of x and r with that of a; 

 and y ; and if, after the time t, r become r+ 

 as, Q, 9+au, and 'sr, ut + ^+av, a being very 

 small, we shall have 



/ddw ^ . „ ^*' \ + 



ar«gfl.(^ — 2/i sm cos d. -5^) ^ 



, . ddv . dw 2^1 sin H \s % 



«r«g4sm^d^+2w sin cosfl^+ — - — .-5^) + 



