412 Attraction of Ellipsoidal Shells and Solid Ellipsoids. 



as an equation in u. When V = 0, the point /, g, h lies on 

 the internal surface of the focaloid. Then (50) agrees 

 with (49). 



34. It is interesting to compare equations (42) and (49) . 

 The first gives the potential produced at airy internal point 

 by a thin elliptic homceoid of mass m, and (49) gives 

 that of a focaloid of mass m at an internal point f\ 

 g, h. If we put f=g — h=zO, we get the potentials at the 

 centre in the two cases. Let it be supposed that both shells 

 are thin, and that both have the same external surface. The 

 thickness of the homoeoid being directly, and that of the 

 focaloid inversely, as the length of the perpendicular let fall 

 from the centre on the tangent plane at the point considered, 

 the potential at the centre must be greater for the focaloid 

 than for the homceoid. The first term on the right of (49) 

 involves for the centre the same integral as does (41) for the 

 homoeoid, but this integral is multiplied by jm' in (49) as 

 against \m in (41). The excess is diminished by the second 

 term in (49) which varies with the deviation of the surface 

 from sphericity ; and also with the thickness of the focaloid 

 on the whole. 



In the particular case in which the surface is spherical the 

 second term just makes the potential at the centre the same 

 for a thin focaloid as for a thin homoeoid, as the reader may 

 verify by evaluation. If we take the case of a solid ellipsoid 

 the second term in (49) vanishes, and the potential at the 

 centre is | of that which would be produced in the interior 

 of a thin homoeoid of the same mass and coincident with the 

 surface of the ellipsoid. 



35. From the expression given in (44) for the potential in 

 the interior of a homoeoid of any thickness, we can readily 

 calculate the work done by gravitational attraction in bringing 

 together from infinite dispersion in space the matter com- 

 posing an ellipsoid, or a homoeoid of finite thickness, whether 

 uniform or made up of homoeoidal shells of different densities. 

 For the case of uniform density, let mass of amount 

 27rpabcli*dli, be brought from infinity to the homoeoid to which 

 (44) refers, and be placed as an additional thin homoeoid 

 on the interior surface. The work done by gravitational 

 attraction in bringing this matter into position is Vm. Hence 



Ym = 27r 2 o 2 a 2 b 2 c 2 hHl-li)dhC — d JL (51) 



Jo _v'(a 2 +w)(A' + «)(c J, + ii) ^ \ 



If then "W denote the whole work done in building up the 



