Ellipsoidal Shells and of Solid Ellipsoids, 393 



where k is less than 1. The equation of the inner surface is 



%- 9 =k—dk, (6) 



as k must diminish from the value unity, for the surface 

 which has the equation 



to zero when the axes are of infinitesimal length. Further, 

 I shall take as the equation of the confocal surface 



2^— =* (7) 



The thickness of the homoeoid at the point A. of co- 

 ordinates x, y, z, on its outer surface is ^pdk/k, and the 

 mass of an element of area d& at the same point of the shell is 

 ^ppdsdk/k, where p is the length of the perpendicular from the 

 centre on the tangent plane at x, y, z, and p is the volume 

 density of the matter of the shell. For the thickness of the 

 shell is clearly 2(«k/a«)/<\/2(**A* 4 ), that is pt(xdxjdP}lk, 

 and by differentiation of (5) this is at once seen to be 

 ^pdk/k. Thus \pdhjk is the constant (3 of § 6 above, and 

 similarly j3 f is found. 



14. 1 shall now establish the geometrical lemma referred to 

 in § 13 above, on which will be based a proof of Poisson's 

 theorem that the resultant force exerted by the homoeoid on a 

 unit particle at the external point P of coordinates /* g, h 

 acts along the normal to the confocal through P : then I 

 shall give a very simple and direct calculation of the amount 

 of this resultant force, and finally obtain the potential of the 

 homoeoid, and of a solid ellipsoid, at any external point. 



Let the enveloping cone be drawn from P as vertex to the 

 external surface (5) of the homoeoid. The points of contact 

 lie in the polar plane of P ; and the internal axis of the 

 cone, the normal at P to the confocal, meets this plane in a 

 point Q. Now let a line drawn from P at any angle O to 

 PQ meet the homoeoid in the two points A, B (see fig. 1). 

 Consider for the present only one of these, A, and let x, y, z 

 be its coordinates, and r denote its distance from P. P is on 

 the confocal represented by (7) : let P x (coordinates / l? g\, lt\) 

 be the corresponding point on the surface (5) of the homoeoid, 

 and A! (coordinates cc\ y\ z') be the point on the confocal 

 corresponding to A. The distance of A' from P x is also r by 



