Ellipsoidal Shells and of Solid Ellipsoids, 411 



This theorem was practically given for points in the axes 

 of an ellipsoid of revolution by Maclaurin. For he states 

 [Fluxions, Art. 650], with the same limitations, the remark- 

 able result that an ellipsoid made np of confocal shells each 

 of uniform density, differing from shell to shell, and an 

 ellipsoid of the same size and of uniform density, exert 

 attractions on an external particle which are in the same 

 direction, and have values in the same ratio as the masses of 

 the ellipsoids. 



The theorem becomes generalized by the extension of 

 Maclaurin' s theorem by Laplace to any form of ellipsoid and 

 any external point. Obviously in the theorem just stated 

 for a heterogeneous ellipsoid, the ellipsoids compared need not 

 be of the same size but only confocal. 



33. If the point/, c/, h considered be in the hollow within 

 the focaloid, the potential can be found by subtracting from 

 the expression for the potential at the point due to the 

 complete ellipsoid, the potential at the same point due to the 

 solid ellipsoid of the same density bounded by the surface 

 %\ar/(a 2 —s)\ = l, the internal surface of the focaloid. 

 Making this calculation by (22), and putting in the result 

 m' (see § 32) for the mass of the focaloid, we get 



V=|m 



■f 



MsS;) 



du 



Jo \/(a 2 + u){b 2 + u)(c 2 + u) 



fo (l-X-^-)du 



J- g <y(a 2 + u)(b 2 + uXc 2 +u) 



where m is the mass of the complete ellipsoid. 



In the remaining case, that in which the point f, </, 7i is 

 within the mass of the focaloid, the procedure is exactly the 

 same as that just described. The form of the result is slightly 

 different : it is 



\ a 1 + u) 



du 



Jo */(a 2 + u)(b 2 + u)(c 2 + u) 



r» (l— 2-y-; — )du 



-i(jn-m')\ ^ a2 + u > (50) 



J a'-. ^ (a 2 +u)(b 2 +u)(c 2 + u) 



where V is the positive root of 2{y 2 /(a 2 — s+u) } = 1, regarded 



