21 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. 



Now the volume of an ellipsoid with axes a, I, c, is f Trabc, that 

 of the inner ellipsoid of the shell 3 is accordingly 



and that of the shell is the increment of this on increasing 6 by 

 dO, or 



(vol. 3) = 



Similarly (vol. 4) = 47r<9 2 d0a 2 /3 2 y 2 . 



Consequently, if we suppose the ellipsoids 1 and 2 filled with 

 matter of uniform density p and p 2 the condition of equal masses 

 of the thin layers 3 and 4, 



is simply 



that is, equality of masses of the two ellipsoids. And since for 

 any two corresponding homceoids such as 3 and 4 (0 and + d6) 

 the attraction on outside points is the same, the attraction of the 

 entire ellipsoids on external points is the same. 



This is Maclaurin's celebrated theorem : Confocal homogeneous 

 solid ellipsoids of equal masses attract external points identically, 

 or the attractions of confocal homogeneous ellipsoids at external 

 points are proportional to their masses.* 



112. Potential of Ellipsoid. The potential due to any 

 homo3oidal layer of semi-axes a, ft, 7, is to be found from our 

 preceding expression for F, 109 (12), 



ds 



where X is the greatest root of 



2 _^_ * 2 



Now if the semi-axes of the solid ellipsoid are a, b, c, those of the 

 shell a = da, /3 = Ob, 7 = 6c, we have M = 4>7r0 2 d0abc, if the density 

 is unity, and 



(i) 



* Maclaurin, A Treatise on Fluxions, 1742. 



