Ellipsoidal Shells and of Solid Ellipsoids. 391 



in which the element of volume is expressed in ordinary polar 

 coordinates. The integral is transformed bv considering a 

 nomothetic shell within the ellipsoid, and taking as the element 

 of volume that intercepted between the two surfaces at an 

 element ds of one of them. Then the integration is effected 

 for the whole shell to which ds belongs, and then carried from 

 shell to shell for the whole ellipsoid. 



It is to be noticed that Poclrigues did not determine the 

 attraction or the potential doe to a single nomothetic shell, 

 but merely imagined the solid ellipsoid divided into such shells 

 in his process of integration. 



10. In the volume of the Memoires de VInstitut (t. xv., 

 1835) already referred to and immediately preceding the 

 memoir cited, is a " Memoire sur l'attraction des Ellipsoides/' 

 also by Chasles, in which the mode of division into nomothetic 

 shells is used, and is attributed to Poisson. There can be no 

 doubt that Poisson was the first to calculate explicitly the 

 attraction of such a shell at an external point, and to apply 

 it to the problem of the solid ellipsoid ; but it is equally clear 

 that the idea of this mode of division is of earlier date. Over 

 this point arose in 1837 a curious reclamation. Another 

 memoir by Chasles, in which the same method was used, was 

 referred by the Academie des Sciences to Libri and Poinsot. 

 The latter reported without mentioning Poisson's memoir of 

 1835, and thereupon Poisson in the Comptes Rendus (t, vi., 

 pp. 83 "-840) called attention to this mode of decomposition 

 of a solid ellipsoid, and affirmed that it offered the only means 

 of reducing the double integrals of the problem to single 

 integrals. Poinsot rejoined re-affirming the priority of 

 Rodrigues in this matter, and the discussion was closed by 

 some further remarks by Poisson, and a second rejoinder from 

 Poinsot. These are to be found at the beginning of the next 

 volume of the Comptes Rendus : a fairly full account of the 

 controversy is given also by Todhunter in his i History.' 



11. In Poisson's memoir of 1835 it is proved that tho 

 resultant attraction of an elliptic homceoid at an external 

 point/, g, h, is directed along the internal axis of the cone 

 drawn from the external point as vertex to touch the homceoid. 

 This is a very remarkable theorem, and attracted very con- 

 siderable attention. For that axis of the cone is the normal 

 to the confocal ellipsoidal surface through /. g, h ; and the 

 theorem at once gives the family of external confocal ellip- 

 soidal surfaces as the equipotential surfaces of the homceoid. 

 The importance of these surfaces was not perceived until later, 

 when the researches of Green, Gauss, Chasles, and others 

 had become known, and had led to new methods of treating 



