On the Attraction of Ellipsoids. 367 



It appears, from this, that the central and transverse components 

 of the attraction of a solid ellipsoid of uniform density, and 

 whose ellipticities are small, on any external point whatever, 

 are given by the same formulae as the corresponding components 

 of the action of any mass on a distant point. 



Now it is a property of moments of inertia, that they are 

 subtractive, that is, the difference of the moments of inertia of 

 two masses with relation to any axis is equal to the moment of 

 inertia of the difference of those masses with relation to the same 

 axis. And the values at which we have arrived for the central 

 force, and for the three components of the transverse force, con- 

 tain in each term either a mass or a moment of inertia in the 

 first power, and therefore, these values also are subtractive. 

 Hence the two components of the attraction of a homogeneous 

 mass contained between two concentric and coaxal ellipsoids of 

 small ellipticities, are given by formulae (10) and (11). Now 

 suppose an ellipsoidal mass to be composed of strata bounded 

 by ellipsoids of different but small ellipticities, each stratum 

 being homogeneous throughout its extent, while the density 

 varies from one stratum to another according to any law ; then, 

 since those formulae hold for the action of each stratum sepa- 

 rately, and since the terms of which they are made up are in 

 their nature additive, they hold for the entire mass.* 



PROPOSITION VIII. 



An oblate Spheroid is composed of spheroidal strata of different 

 densities and of variable but small ellipticities ; to find the Com- 

 ponents of its Attraction on any external point. 



The expressions given in the last Proposition for R and P 

 become simplified in this case. Let OZ be the axis of revolu- 

 tion, and let A denote the angle which OM makes with the plane 



* See Professor Mac Cullagh, in the " Dublin University Examination Papers," 

 1833, p. 268. 



