368 On the Attraction of Ellipsoids. 



XY ; then since A and B are equal, we have 



I = A cos 2 A + C sin 2 A ; 

 and therefore 



A + B + C-M = (C -A) (1-3 sin 2 A); 

 also 



M OS x OT = (A - C} sinA cos A. 



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

 have 



*-S+<l-Brin'A), (13) 



P = 3 - 4 - cos A sin A. (14) 



The direction of the force P is towards the plane of the 

 equator ; this appears from the shape of the " ellipsoid of gyra- 

 tion," which in this case is a prolate surface of revolution. 



PROPOSITION IX. CLAIRAUT'S THEOREM. 



Whatever be the Jaw of variation of the Earth's density at 

 different distances from the centre, if the ellipticity of the surface be 

 added to the ratio which the excess of the polar above the equatorial 



K 



gravity bears to the equatorial gravity, their sum will be ^ q, wAere 



<e 



q is the ratio of the centrifugal force at Equator to equatorial gravity. 



For suppose the attracted point M to be on the surface of 

 the earth, which is known to be an oblate spheroid of small 

 ellipticity. Then, from the principles of Hydrostatics, since the 

 tangential force is zero, we have 



R cos 6 - P sin 9 - o> 2 r cos A cos (0 - A) = 0, (15) 



where w denotes the angular velocity, and the angle which 

 the tangent to the meridian through the attracted point makes 

 with the radius vector. Developing cos (0 - A) and arranging, 

 we obtain 



(R-w*r cos 2 A) cos = (P + w 2 r cos A sin A) sin 0. (16) 



