FIGURE OF A GRAVITATING BODY. ^Jl 



cisely in this ratio, the matter which produces it being al- 

 ways similarly arranged, and varying only in quantity; con- 

 sequently the sum of this attraction and the original disturb- 

 ing force both vary as the inclination of the surface, and 

 may be in equilibrium with the tendency to descend towards 

 the centre, provided that the elliptit ity be duly commensu- 

 rate to the density of the elevated parts. 



In the last place we must investigate what is the mag-ni- f*^ag"itut5e of 



*■ J c 4.1 ^^■ ^ ■ 1- . ,. the ellipticitT, 



tude ot the elhpticity corresponding to a given disturbing 

 force and a given depsity. It follows from the proportions 

 already mentioned, first, that the effectual elevation at each 

 point of each concentric semicircle is proportional to the 

 sine of its distance from the bisecting plane; and secondly, 

 that the greatest eftective elevation of each semicircle, for 

 any one posiiiou of the supeificial particle, is as the product 

 of the sine and the cosine of the angular distance from that 

 particle, the diameter of the circle being as the sine, and 

 the distance of its centre from the equatorial plane as the 

 cosine. It may easily be shown, that the disturbing force, 

 reduced to the direction of the surface, or of the plane of 

 each circle, is equal to the attraction which would be exerted 

 by the matter covering the whole semicircle to a height equal 

 to half the greatest elevation, if placed at the middle point: 

 for the elevation being as the sine of the distance from the 

 bisecting plane, and the comparative effect being also as the 

 sine, the attraction for each equal particle of the semicircle 

 is as the square of the sine, and the whole sum half as great 

 as if each particle produced an equal effect with that on 

 which the elevation is greatest. We must therefore com- 

 pute the attraction of the quantity of matter thus deter- 

 mined, supposing it to be disposed at the respective points of 

 a great circle passing through the given point and the pole 

 of the spheroid. The immediate attraction of each particle 

 being inversely as the square of the chord, its effect reduced 

 to the common direction will be as the sine directly, and the 

 cube of the chord inversely, and this ratio being com- 

 pounded with that of the product of the cosine and the 

 square of the sine, which expresses the quantity of matter 

 at each point, the comparative effect will be as the cube of 

 the sine and the cosine directly, and as the cube of the 

 P 2 chgrd 



