380 



Mr. George J. Allman's Account of 



Proposition I. 



IfP he any point on the surface of an ellijysoid, and PCi be drawn perpen- 

 dicular to an axis OC, and an ellipsoid he described through Ci concentric^ similar 

 and similarly placed to the given ellipsoid ; then the component of the attraction of 

 the given ellipsoid on P in a direction parallel to OC is equal to the attraction of 

 the inner ellipsoid on the point Ci. 



This theorem is an ex- 

 teusioa of that given by 

 Mac Laurin* relating to 

 the attraction of a sphe- 

 roid on a point placed on 

 its surface. It may, more- 

 over, be established by 

 means of the same geo- 

 metrical proposition from 

 which Mac Laurin de- 

 duced his theorem. 



Through the point P 

 let a chord PP' of the given ellipsoid be drawn parallel to the axis OC ; now 

 suppose both ellipsoids to be divided into wedges by planes parallel to each 

 other, and passing respectively through this chord and the parallel axis of 

 the inner ; and suppose the wedges to be divided into pyramids, the common 

 vertex of one set being at P, and of the other at Ci . Observing that any two 

 of these parallel planes cut the two surfaces in simdar ellipses, such that the 

 semi-axis of one is equal to the parallel ordinate of the other, it is easy to see 

 that the reasoning employed by Mac Laurin may be used to establish the 

 truth of the theorem stated above. 



Fig. 1. 



* De Caus. Phys. Flux, et Refl. Maris, sect. 3. Or sea Airy's Tract on the Figure of tlie 

 Earth, Prop. 8. 



