Direct attraction of a spheroid. qj^ 



lY. 



Calculution of the direct Attraction of a Spheroid, and De^ 

 monstration o/Clairaut's Theorem. By a Correspondent. 



To Mr. NICHOLSON. 

 SIR, 



-J- HE same moJe of calculation, by which the figure of Extension of 

 a gravitatins^ body, differing but little from a sphere, has ggaSher' 



been determined (p. 208 of this volume), is also applicable 

 to the magnitude of its immediate attraction, or the com- 

 parative length of a pendulum in different latitudes. 



Suppose a sphere to be inscribed in the spheroid, and Attraction of 

 another to be circumscribed about it : I shall first show, ^^^ promiaent 

 that the attraction at the pole is equal to that of the smaller 

 sphere increased by yV of that of the shell, and at the equa- 

 tor equal to that of the larger diminished by -j-V* If we call 

 the attraction of this shell 2, its surface being equal to the 

 curved sxirface of a circumscribing cylinder, the attraction 

 of a narrow ring of this cylinder, or of the elevated portion 

 of the spheroid at the equator, supposed to act at the dis- 

 tance of the radius, or unity, may be expressed by its 

 breadth : but in its actual situation its attraction in the di- 

 rection of the axis is reduced in the ratio of the cube of the 

 chord of half a right angle to the cube of the radius ; and 

 the attraction of any other ring will be to this in the ratio 

 of the quantity of matter, or the cube of the e.'ue of the 

 distance from the pole, and of the versed sine directly, and 

 in the ratio of the cube of the chord inversely; that is in 

 the joint ratio of the cube of the cosine of half the angle 

 and the versed sine : thus, if we call the cosine of half the 

 angle.x, the versed sine being 2 — 2 a:*, and the fluxion of 



2 X 



the arc -, the fluxion of the force at the equator 



V {l-xxy 



^iu Uq « . — ^ , and elsewhere as much less as 



V«i.. XX^Aus. 1308. T »* 



