414 



VIII. NEWTONIAN POTENTIAL FUNCTION. 



external points is the same, or if the masses differ, is proportional 

 simply to the masses of the layers. For it depends only on A, 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 homceoid. 

 The theorem was given in this form by Chasles. 1 ) 



155. Maclaurin's Theorem. Consider two confocal ellipsoids, 1, 

 Fig. 142, with semi -axes e^, @ lt y lf and 2, with semi -axes 2 , /3 2 , y 2 . 

 The condition of confocality is 



16) 



Fig. 142. 



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 

 wcc-i* v'Lj-i, vV-\ and 01 4 are / ucc^ i/p^. v^o. 

 and hence the condition of confocality, 



17) 



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

 layer between 3 and another ellipsoid for which # is increased by d&, 

 and on 4 a homoeoidal layer given by the same values of & and d&, 

 and furthermore choose the densities such that these two homoeoidal 

 layers have the same mass, then (since these homceoids are confocal) 

 their attractions at external points will be identical. 



Now 



the volume of an ellipsoid with axes a, 6, c, is stabc, 



o 



that of the inner ellipsoid of the shell 3 is accordingly 



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

 (vol. 3) = 



(vol. 4) = 



or 



Similarly 



1) Chasles, "Nouvelle solution du probleme de 1'attraction d'un ellipsoide 

 h^terogene sur un point exterieur. Journal de Liouville, t. V. 1840. 



