264 The Eev. Samuel Haughton on the 



^ 47rJ/)rt- 47ra r d.a^e 



r 5r' 



in which p is the density of any couche, a the radius of its equi-capacious sphere, 

 and e its ellipticity. 



The potential of a sliell composed of conches arranged in the manner sup- 

 posed, on an internal point, is, 



F=M^-l^>*. (.0, 



The radius vector of the siurface of each couche is given by the following equation, 



r = a(l-f»); (11) 



from which may be deduced the values of the equatorial and polar axes, viz.^ 

 a (1 + ^e), and a (1 - |^). Substituting from the foregoing equations in (7), 

 we find 



^onst = - (I + e.)\ pa-— ..^p- 



d.w'e 



da 



(* 47ra= f*. de 4^^ f» 



a denoting the mean radius of the external si;rface, ai the mean radius of the 

 internal surface of the shell supposed solid, and m the ratio of centrifugal force 

 to gravity at the equator. This equation consists of two parts, one independent 

 of s, which is satisfied by means of the constant ; the second, which is the coef- 

 ficient of », gives the condition, 



er" 2 If" d.a^e a^ (^ de ma? ^ „_ ^. 



This equation expresses the fact, that each fluid surface is perpendicular to the 

 resultant of all the forces acting upon the particles composing it. 



Differentiating this equation, so as to banish the integrals, we obtain, 



Jo Ju 



