[ 203 ] 



XXXV. On the Attraction of an Homogeneous Ellipsoid on 

 an external Particle. By J. B.* 



T ET a, h, c be the semiaxes of the eUipsoid, a the greatest 

 -*-' and h the least, 8 = distance of external particle from the 

 centre, assumed as the origin of coordinates, A, |x, v the angles 

 which the line 8 makes with the semiaxes a, 6, c, M = mass 

 of the ellipsoid, jx the external particle, andy the intensity of 

 attraction at the unit of distance, D the attraction along the 



line 8 ; then in the expression V = 1 1 f—^y expanding -k-> 



multiplying by dm^ integrating and including the terms as far 

 as the third powers of the coordinates of dm, and making the 

 necessary reductions, we have this general formula : 



j^_f/^M f , L 3 ^(3 cos'^A— 1 ) a^+(3 cosa ^— l)&^-f (3 cos'^u- 1) c % ) 



2« Let A, B, C be the attractions on three particles at 

 equal distances from the centre in the axes produced; then 



I' 



{^^l^-^^)} 



B..^^^{,,-|(^^!r^)}; 



./>M 



8« 



Adding these three formulae, A + B + C = ^^ — , or the 



sum of the attractions of an ellipsoid on three particles at 

 equal distances from the centre in the axes produced = 

 three times the spherical attraction of the same mass on a 

 particle at the same distance. 



3. Hence, also, the attraction of the ellipsoid is greatest in 

 the direction of the greater axis, and greater than that of a 

 sphere of the same mass on an equidistant particle ; and that 

 in the direction of the lesser axis is the least. 



4. If the squares of the semiaxes are in arithmetical pro- 

 gression, the attraction of the ellipsoid on a particle situated in 

 the mean axis = to that of a sphere of the same mass on a 

 particle at the same distance, for (2c'— a*— 6*) in this case 

 = 0. 



5. Let there be two homogeneous concentric ellipsoids of 



* Communicated by the Author. 

 2D2 



