212 FIGURE OF A GRAVITATING BODY. 



chord inversely, or as the cube of the cosine of half the arc 

 and the cosine of the whole arc conjointly. If therefore 

 we call the cosine of half the arc x, the cosine of the whole 

 arc will be 2 x^ — 1, and the fluxion of the arc being 



that of the force will be 



• {!—:*: x)' ^""-"' ^ "" /{I— XX) ' 



of which the fluent is (|^ x* + | x* + ■!-) / (l — xx), as 

 may be shown by substituting, in the reduction of its 



fluxion, • for >/ (l — xx): and while x decreases 



/ (1 — xx) 



from 1 to 0, this fluent becomes ^. But in order to deter- 

 mine the unit with which this quantity is to be compared, 

 we must consider the initial force as unity, and imagine, 

 that it is continued through an arc equal in length to the 

 radius; and we must find the attraction of the solid con- 

 tained between a circular plane and a conical surface, ini- 

 tially touching the effective portion of the elevation, and 

 including it between them; the attraction reduced to a 

 common direction, being initially half the whole attractive 

 force of such a solid, as we have already seen of the con- 

 centric circles considered separately. But the attraction of 

 any slender conical or pyramidical body for a particle 

 placed at its vertex, is three times as great as that of the 

 same quantity of matter situated at its base; consequently 

 the attraction of the supposed solid is equal to that of the 

 circumscribing semicylinder placed at the distance of the 

 radius : the conical excavation being half of the solid, and 

 the semicylinder triple of the cone: but the height of this 

 semicylinder in the case of a particle situated half way be- 

 tween the pole and the equator of the spheroid, is twice the 

 ellipticity, the tangent of the angle of mutual inclination 

 of tlie surfaces of the etfective elevation being initially equal 

 to twice the greatest ordinate, because the product of the 

 sine and cosine, when greatest, is equal to half of the ra- 

 dius : the semicylinder will therefore be equal to a cylinder 

 of which the diameter is equal to that of the sphere, and 

 the height equal to the ellipticity; and the contents of thi* 

 cylinder will be to that of the sphere, as |- of the elHpticity 

 to the radius. Such therefore is the unit with which the 

 disturbing attraction is to be compared; and when the den- 

 sities 



