154, 155, 156] ATTRACTION OF CONFOCAL ELLIPSOIDS. 415 



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

 of uniform density ^ and p 2 , the condition of equal masses of the 

 thin layers 3 and 4, 



is simply 

 18) 



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

 two corresponding homoeoids such as 3 and 4 (# and # -f dti) 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. 1 ) 



156. Potential of Ellipsoid. The potential due to any 

 homoeoidal layer of semi -axes a, /3, y is found to be from our preced- 

 ing expression for F, 14), 



14) 



M r 



* V vw 



where A is the greatest root of 



^^ 



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

 the shell a = &a, /3 = #&, y = frc, we have M = k-st^d&abc, if the 

 density is unity, and 



20) d,V **9***dbcJ- 



where A is defined by 

 21) 



To get the potential of the whole ellipsoid, we must integrate 

 for all the shells, and 



22) V^Zxalfi fad ft C ds 



J J V(aW + 



1) Maclaurin, A Treatise on Fluxions. 1742. 



