109 111] 



ATTRACTION OF ELLIPSOIDS. 



209 



are ellipsoids confocal with it. Consequently if we distribute 

 on any one of a set of confocal ellipsoids a layer of given 

 mass whose surface density is proportional to 8, the attraction of 

 such various layers at given external points is the same, or if 

 the masses differ, is proportional simply to the masses of the layers. 

 For it depends only on X, which depends only on the position of 

 the point where we calculate the potential. 



Since by the definition of a homceoid, the normal thickness of 

 an infinitely thin homceoid is proportional at any point to the 

 perpendicular on the tangent plane, we may replace the words 

 surface layer, etc., above by the words homogeneous infinitely thin 

 homoeoid. The theorem was given in this form by Chasles.* 



111. Maclaurin's Theorem. Consider two confocal ellip- 

 soids, 1, Fig. 50, with semi-axes i, &, 71, and 2, with semi-axes 



FIG. 50. 



a 2> /5 2 , 72- The condition of confocality is 



- 



= 2 - 2 = 2 - 



= 72 - 7l = s, say. 



If we now construct two ellipsoids 3 and 4 similar respectively to 

 1 and 2, and whose axes are in the same ratio to those of 1 and 

 3, these two ellipsoids 3 and 4 are confocal (with each other, 

 though not with 1 and 2). For the semi-axes of 3 are 0a lt 0/3 1} 6y lf 

 and of 4 are # 2 , #/3 2 , #72, and hence the condition of confocality, 



#V - 0V = 2 /3 2 2 - 6% 2 = 6V - 0V = fr s 



is satisfied. Now if on 3 we distribute one infinitely thin homoeoidal 

 layer between 3 and another ellipsoid for which is increased by 

 dO, and on 4 a homoeoidal layer given by the same values of 6 and 

 d&, and furthermore choose the densities such that these two 

 homceoidal layers have the same mass, then (since these homceoids 

 are confocal) their attractions at external points will be identical. 



* Chasles, "Nouvelle solution du probl^me de 1'attraction d'un ellipsoide 

 h6tdrogne sur un point extrieur." Journal de Liouville, t. v. 1840. 



W. E. 14 



