392 Prof. A. Gray on the Attraction of 



problems of attraction — methods which had become essential 

 for the progress of the theory of electricity. 



Poisson resolved the force due to each element of the 

 homoeoid along the axis of the cone, and by expressing the 

 component in terms of polar coordinates referred to/, g, h 

 as origin, was able to obtain the resultant force in an integrable 

 form. His process is one of direct integration of the 

 expressions obtained, and involves some troublesome con- 

 siderations as to the limits of the integration with respect to 

 #, the angle between the axis of the cone and the line drawn 

 to the element considered, and therefore runs to considerable 

 length. 



12. A very different process of calculating this integral is 

 followed by Chasles in his memoir already cited, " Sur Pattrac- 

 tion d'une couche ellipsoidale/'' There the theorem of Lame 

 (given for the steady motion of heat in a uniform solid), 

 that Laplace's differential equation of the potential is 

 integrable when the equipotential surfaces are known, is 

 employed for the family of equipotential surfaces revealed 

 by Poisson's theorem, and the attraction is reduced to the 

 evaluation of a constant left undetermined by the integration 

 of the specialized form of the differential equation for this 

 case. This is effected by considering the particular case of 

 the attracted point on the surface, evaluating the integral for 

 this case, and comparing w 7 ith the expression obtained from 

 the integration of the differential equation. 



13. I shall now show how the integral for the resultant force 

 at/, g, h, expressed in terms of the element c/s, its coordinates 

 x, y, z, the perpendicular p from the centre on the tangent 

 plane at ds, the distance r from the point /, g, It to ds, and 

 the angle between this line and the axis of the cone, can, 

 by means of a simple geometrical theorem of confocal surfaces, 

 which I have not before seen remarked, be transformed to an 

 immediately integrable form, so that the whole calculation 

 can be set forth very briefly. 



In order that the result may be at once applicable to the 

 calculation of the potential of a solid ellipsoid, I take as the 

 equation of the outer surface 



o '> 9. 



a~ c 



or in the usual abridged notation 



Sr,=* ( 5 ) 



t5 v 



x" 



