FIGURE OF A GRAVITATING BODY. 209 



to remove the body from this plane. A second disturbing 

 force will also arise from the want of parallelism in the direc- 

 tion of the attractive force, which will tend towards the line 

 joining the centres, and will be every where to the whole 

 force as the distance from this line to the distance of the 

 bodies. Now if each of these forces be reduced to the di- 

 rection of the circnmference of the sphere, from which the 

 figure is supposed to vary but veiy little, it will be every 

 where proportional to the product of the sine and cosine 

 of the distance from the equatorial plane, and when this 

 distance is half a right angle, each of them will be half as 

 great as in its intire state. Thus the gravitation towards the 

 moon at the earth's surface is to the gravitation towards the 

 earth as 1 to 70 times the square of 60j, or to 256217, and 

 the first disturbing force is to the whole of this as 2 to Uo|, 

 at the point nearest to the moon, and the second as j to Cof 

 at the equatorial plane ; and the sum of both reduced to 

 the direction of the circumference where greatest, as 3 to 121, 

 that is, to the whole force of the earth's gravitation as 1 to 

 10,334,000. And in a similar manner the joint disturbing 

 force of the sun is to the weight as 1 to 25,736,000. 



Now if a sphere be inscribed in an oblong spheroid, the Inclination of 

 elevation of the spheroid above the sphere must obviously be ^^'^I^'^qj^j 

 proportional, if measured in a direction parallel to the axis 

 of the spheroid, to the ordinate of the sphere, that is, to the 

 sine of the distance from its equator ; and if reduced to a di- 

 rection perpendicular to the surface of the sphere, it must 

 be proportional to the square of that sine; and the tangent of 

 the inclination to the surface of the sphere, which is as the 

 fluxion of the elevation divided by that of the circumference, 

 must be expressed by twice the continual product of the 

 sine, the cosine, and the ellipticity or greatest elevation, the 

 radius being considered as unity: so that the ellipticity will 

 also express the tangent of the inclination where it is great- 

 est ; and the inclination will be every where as the product 

 of the sine and cosine. 



If therefore the density of the elevated parts be considered Tidcsofaspaot 

 as evanescent and their attraction be neglected, there will be inconsiderable 

 an equilibrium when the ellipticity is to the radius as the 

 disturbing force to the whole force of gravitation : for each 



Vol. XX— June, 1808, V particle 



