388 Prof. A. Gray on the Attraction of 



components are given by (1), and thus the so-called external 

 problem is reduced to the internal problem of which the 

 solution is known. 



5. The external problem was, however, solved directly by 

 Poisson in the memoir above referred to, by the device, which 

 he appears to have been the first to adopt, of imagining the 

 ellipsoid divided into infinitely thin similar and similarly 

 situated ellipsoidal shells, or " elliptic homceoids " as they 

 have been called by Thomson and Tait**, then finding by 

 direct integration the attraction exerted by such a shell on 

 a unit particle at the given external point, and finally passing, 

 by another integration from shell to shell, to the attraction 

 exerted by the solid ellipsoid. 



[The term nomothetic ellipsoidal shell is used by many 

 writers for the " couche elliptique " on which Poisson based 

 his solution, and is perhaps less open than the term elliptic 

 homceoid to objection on the ground of derivation ; but we 

 shall adopt the name elliptic homoeoid, or simply homoeoid, 

 where there is no risk of ambiguity. Thomson and Tait also 

 give the name " focaloid " to a shell bounded by two confocal 

 ellipsoidal surfaces.] 



6. On Ivory's notion of corresponding points Chasles based 

 a very important and elegant theorem of the attraction of 

 confocal homoeoids. Imagine two elliptic homoeoids, A, B, 

 of infinitesimal thickness, and each of uniform (not necessarily 

 the same) density, the two outer surfaces and the two inner 

 surfaces of which are confocal. Let a, b, c be the lengths of 

 the semi-axes of A, and a', //, c f the same quantities for B. 

 Also let P and Q be corresponding points on the two shells. 

 The theorem of Chasles affirms that the potential at the point 

 Q due to the homoeoid A is to the potential at P due to the 

 homoeoid B, as the mass of A is to the mass of B. 



For let rfs be an element of A and dW the corresponding- 

 element of B, and p, p' the length of the perpendiculars from 

 the centre on the tangent plane at the elements. It is easy 

 to prove that 



p'ds' _ a'h'c' ,o\ 



pds abc 



The masses of corresponding elements of the shell are fipds. 

 fi'p'ds 1 , where /3, @ f are constants depending on the density and 

 scale of thickness in the two cases. The total masses are Air/Sabc, 

 47T/3V6V respectively. Hence it follows that the masses of 

 corresponding elements are in the ratio of the total masses in 

 the two cases. [See § 13 below for the value of J3, /3'.] 



* Nat. Phil. Part II. § 494 g, 



