Ellipsoidal Shells and of Solid Ellipsoids. 387 



point P on the surface, and similarly a', &', c', %', y\ .:' the 

 lengths of the semi-axes of the larger ellipsoid, and the co- 

 ordinates of a point Q upon it. If the coordinates fulfil 

 the relations x/a = x'/a\ y/b=y'/b\ z/c = z r /c\ the points P. 

 Q are corresponding points. It is easy to show that if 

 P., P' be two points on the first ellipsoid, Q, Q' the corre- 

 sponding points on the other, and the ellipsoids be confocal, 

 the distances PQ', P'Q are equal. 



Xow considering attractions in the direction of the principal 

 axes, and taking first the axes «, a\ it can be proved that the 

 attraction X of the first ellipsoid, A say, on a particle of unit 

 mass at Q on the surface of the other ellipsoid, B, is to the 

 attraction X / of the ellipsoid B on a unit particle at the 

 corresponding point P on the surface of A, in the ratio bcjb'c' . 

 Similarly Y/Y'=m/cV 5 Z/Z' = ab/a'l/. The ellipsoids are 

 here supposed to be solid and of uniform density, p say, and 

 confocal. If we call the masses of the ellipsoids M, M', then 

 since M/M f ==a&c/a'6V, we may express the theorem in the 

 form (not given by Ivory) 



X _ c/M_ Y _ //M Z _ r/M 



X' ~ a M' ' Y' ~ // M" Z'~ cW ' ' W 



TThen expressed in this form the theorem is evidently 

 true whether the two ellipsoids have the same density or not, 

 provided that each is homogeneous. For the components of 

 force on a unit particle evidently vary with the masses of 

 the ellipsoids, when their dimensions remain unchanged, 

 and therefore a change in the ratio M/M', caused by varying 

 the density of either ellipsoid, is represented by a correspond- 

 ing change in each of the ratios X/X', Y/Y', Z/Z'. 



In the particular case of M=M', the theorem takes the 

 form 



X' ~ a ' Y' ~ b ' 7J ~~ e ' ' m ' W 



It was first pointed out by Poisson that Ivory's theorem is 

 true for every law of attraction^ provided the law is a function 

 of the distance only. 



4. Let us now suppose that the problem of finding the 

 attraction of a homogeneous ellipsoid at an internal point has 

 been solved, and that it is required to find the attraction at 

 an external point, Q say. It is only necessary to find for 

 the corresponding point P on the surface of the given ellip- 

 soid the attraction exerted on a unit particle by the confocal 

 ellipsoid, the surface of which contains the point Q. The 



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