On the Attraction of Ellipsoids. 365 



PROPOSITION YIL 



An Ellipsoid is composed of ellipsoidal strata of different densities 

 and of variable but small ellipticities ; to find the Components, cen- 

 tral and transverse, of its Attraction on an external point. 



The values found in the last Proposition for the components 

 of the attraction of any mass on a very distant point will be 

 found to hold in the present case, whatever be the position of 

 the attracted point. In order to show this, we shall first prove 

 it for a homogeneous ellipsoid of small ellipticities. Such an 

 ellipsoid being given, another confocal with it can be con- 

 structed so small, that the distance to the attracted point may be 



but from (7), 



x'Y- y'X = -- - cos a cos ft' = -- - (a 2 - i 4 ) cos a" cos ft', 



q / T> _ f~1\ "\M 



y' Z z'Y = -- - - - cos ft' cos 7' = -- - (b* - c 2 ) cos ft' cos 7', 



3(0 -A) 3M.. 



z X - x Z = -- - cos 7 cos a = -- TJJ- (o 2 a 2 ) cos 7 cos a . 



Now it is well known, that 



( 2 - 2 ) cos o' cos ft', | (i 2 - c 2 ) cos ft' cos 7', i (<? 2 - a 2 ) cos 7' cos a, 



are the areas of the projections of the triangle OST on the principal planes. Hence 

 it follows that the resultant moment lies in the plane of the radius vector OT and 

 the perpendicular OS to the corresopnding tangent plane of the ellipsoid of gyra- 

 tion ; the tangent plane being perpendicular to OM. It appears, also, that the 

 magnitude of the resultant moment is 



and therefore that the transverse component of the attraction 



. 



Or, the values of the central force and the moment of the transverse force may 

 be obtained directly from the expression (6) for the potential T. This function is 

 of such a nature, that its* differential coefficient with relation to any line (the sign 

 being changed) is equal to the resolved part of the attraction in that direction ; and 

 the differential coefficient with relation to any angle (the sign being changed as 



