Professor MacCullagu's Lectures on the Attraction of Ellipsoids. 393 



on any external point whatever, are given by the same formulte as the corre- 

 sponding components of the action of any mass on a distant point. 



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

 that is, tlie 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, contain in each term either the mass or a moment of inertia in the first 

 power, and, therefore, these values also are subtractivc. Hence the two com- 

 ponents of the attraction of a homogeneous mass contained between two con- 

 centric and coaxal ellipsoids of small ellipticities, are given by formula (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 formulse hold for the action of each 

 stratum separately, and since the terms of which they are made up ai'e 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 ; find the components of its attraction on any 

 external point. 



The expressions given in the last Proposition for R and P become simpli- 

 fied in this case. Let OZ be the axis of revolution, and let \ denote the angle 

 which OM makes with the plane XY; then since A and B are equal, we have 



I=A cos^X-f Csin^X, 

 and therefore, 



A+B+C-?,I={C-A) (l-3sin^X), 

 also, 



IT (OS X ST) = (.4 - C) sin X cos X. 



Substituting these values in the expressions for R and P, we have 



* See Professor Mac CuLLAGH, in the University Examination Papers, 1833, p. 268. 



