335 



agj and outwardly by the spheroidal surface with the mean 

 radius ai. 



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

 the spheroidal surface having the mean radius ai, and out- 

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



" (4.) Centrifugal force. 



" If Uo, Ui, 11-2, &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, 



^•'^"^^ _±_dHJi ^rUi^Q^ 



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,* 



(1) 



3r 



47r 



//,,3_^.,3^^), 4-_ 



3r 



(a) 



(3) |^(«-«.^)4{-<^^^^-^3^3.&c.) 



-«xai^(gc/. + f;;3C73 + &c.)} 



(4) yr^-\gr'{<^o^^B-l)' 



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 Analytique du Systeme du Monde, livre v. 

 No. 32. 

 t Ibid. 



