VOL. XL.J PHILOSOPHICAL TRANSACTIONS. 20Q 



which may be supposed because of the smaUness of nm; we shall have 

 c* X PM* X P/> X — J for the attraction of any one of these elements ; put- 

 ting c for the ratio of the circumference to the radius, and «-for the given 

 ratio of mn to pm, that is, of de to cd. 



Now if we make ca =: e, cb = r, am = z; and for pm, ap, p/», if we sub- 

 stitute their values expressed by z, and then seek the fluent of the foregoing 



quantity ; we shall have — —^ for the value of the whole attraction of 



the solid generated by the revolution of BDbsB : to which if we add -- — , the 



attraction of the sphere, we shall have — 1 — ^^ for the required 



attraction of the spheroid on the corpuscle a. 



Problem II. Supposing now tfie Spheroid Bsbe, fig. 10, to be no longer of a 

 homogeneous Matter, but to be composed of an infinite Number of Elliptical 

 Strata, all similar to BEb, the Densities of which are represented by the Ordi- 

 nates kt of any Curve ivhatever vx, oj which we have the Equation between 

 CK and KT ; the Attraction is required tvhich this Spheroid exerts on a Corpuscle 

 placed at the Pole b. — 2. Making bc ^ e, CK = r, by the foregoing proposi- 

 tion, we should have — 1 — -— — - for the attraction of the spheroid 



KLK, if it consisted of homogeneous matter ; and the fluxion of this quantity 

 1 — would be the element or moment of the orb klk^M. 



But because the density is variable, we must multiply this value of the attrac- 

 tion of the orb by kt, and tiie fluent of this quantity will be the value of the 

 attraction of the spheroid klk. 



As to the value of kt, which expresses the density of the stratum KLKklk, 

 we shall take only fr'' -\- g?-^, because we shall see afterwards, that a value 

 more compounded, as _/r* -)- g-;' -}- hr' + ir', &c. which by the property of 

 series may express all curves, would not produce any variety in the calculation. 



Therefore multiplying the foregoing equation by fr -\- gr^, we shall have 



2cfxl + 2»xr^'^f _ ic»fr ^'''^ , 2c gx l + 2«xr ^+? _ 4c«gr^ + ? r ., .. r 



eex3+p e*x5+p eex3 + q e^x5 + q 



attraction of the spheroid klk, exerted on a corpuscle placed at b. 

 3. In this value making r = e, we shall have 



~ h =^ h -■ + -L. ■ which will express the force of 



3+p 3+PX5 + P 3 + q 3+JX5 + 9 '^ 



attraction of the spheroid be6, exerted on a corpuscle placed at the pole b. 

 Theorem. ^ Corpuscle being placed in any Point n of the Surface of the 

 VOL. vm. E E \ 



