386 Prof. A. Gray on the Attraction of 



and others; but perhaps this solution of the older and simpler 

 problem may not be without interest. 



2. .No problem in the Theory of Attractions has received 

 more attention than that of the attraction of a solid ellipsoid 

 on an external particle. The subject engaged the attention 

 of Newton and Maclaurin*, who dealt with it by a combination 

 of the geometrical methods of which these mathematicians 

 had such a perfect mastery, and the results of Newton's own 

 calculus of fluxions and fluents (differential and integral 

 calculus) ; and all the celebrated analysts of the end of the 

 eighteenth century and the beginning of the nineteenth, 

 Laplace, Lagrange, Ivory, Gauss, Poisson, Chasles, wrote 

 memoirs on the subject which have become classical. To 

 find the attraction of the ellipsoid at an internal point had 

 been soon found to be a comparatively easy matter. The 

 process adopted was to take tlie attracted point as origin of 

 polar coordinates by which the positions of the particles of the 

 ellipsoid (supposed of uniform density) were specified, to 

 express the volume of an element of the attracting body by 

 these coordinates, and then the components of attraction 

 on the particle as triple integrals with respect to the radius- 

 vector and the two angular coordinates. As Poisson puts it 

 in the introduction to the very remarkable memoir f read to 

 the Academie des Sciences on October 3, 1833 :— u The 

 integration with reference to the radius-vector can be carried 

 out without any difficulty, and in the case of an internal 

 point a second integration is easily effected, so that the three 

 components of attraction are expressed by single integrals, 

 reducible to elliptic integrals of the first and second kinds. 

 These integrals are obtainable in a finite form when the 

 ellipsoid is one of revolution. When, however, the attracted 

 particle lies outside the ellipsoid, the double integrals contain 

 a radical, and have limits which render them much more 

 complicated, so that, instead of carrying out the second 

 integration directly, we have to turn the difficulty by reducing 

 the problem for the external point to that for an internal* 

 point, which leaves the problem of the direct integration 

 unsolved. For this reduction the theorem of Ivory leaves 

 nothing to be desired." 



3. Ivory's theorem depends on his notion of corresponding- 

 points on two ellipsoidal surfaces, the axes of which are 

 coincident. Let a, b, c be the lengths of the semi-axes of 

 one, say the smaller, ellipsoid, a:, y, z the coordinates of a 



* See Newton's Priticipia, Lib. I., ss. xii. and xiii., and Maclaurin's 

 ' Treatise on Fluxions,' vol. ii. 



f MSmoires de V Academie Hoy ale des Sciences de VInstitut, t. xiii., 1835. 



