335 



ai, and outwardly by the spheroidal surface with the mean 

 radius oi. 



«' (3.) Attraction of the mass of fluid bounded inwardly by 

 the spheroidal surface having the mean radius «i, and out- 

 wardly by the spheroidal surface, having the mean radius a. 



*' (4.) Centrifugal force. 



" If t/o, [/i, Uii &c., represent such functions of the co- 

 ordinates of the spheroid, that on the substitution of each 

 successively for Ui in the following well-known differential 

 equation it will be satisfied, 



sinddO sin-O dw^ dr' 



and if we use the notation of M. de Pontecoulant for all quan- 

 tities not otherwise specified, we shall have, for the functions 

 on which the forces above enumerated depend,* 



4Tra2^D 1 



(1) 



3r 



471 



3r 



Ka) 



(2) l^,«,_,3)Z)..^!r^(|;!^..«^3^3.&C.) 



(3) lj(a3-a.3).il{aa3(^,Y. + ^r3.&c.) 



-a.a.3(^^C7, + ^3f73 + &c.)} 



(4) yr^^-^gr^{co^^e-^). 



Uo, Uu Yo, Yi, being omitted by the properties of such 

 functions,! and ai being a small quantity depending on the 

 ellipticity of the spheroidal surface bounding the solid mass. 

 The equation of equilibrium of the fluid surface will therefore 

 be 



• Pontecoulant, Theorie Analytiquc du Systeme du Monde, livre v. 

 No. 32. 

 t Ibiil. 



