'210 PHILOSOPHICAL TRANSACTIONS. [anNO 1738. 



foregoing Spheroid BEbe, / say it will undergo the same Attraction from this 

 Spheroid, as if it were placed at the Pole N of a second Spheroid revolving about 

 the A.ve no, the second Axe being the Radius of a Circle equal in Superficies 

 to the Ellipsis fg ; supposing this second Spheroid ngof (fig. 11) to be com- 

 posed of the Strata Minqa, whose Densities are the same as those of the Strata 

 KkLlKk, of the first Spheroid. — 4, In the discourse I had the honour of com- 

 municating to the R. S. being then at Toriieo, printed in the Philos. Trans. 

 N° 445, I have demonstrated this proposition as to a homogeneous spheroid ; 

 and the same reasoning will obtain in this case also. 



Problem. III. To find the Attraction which the Spheroid BEbe (fig. 10) 

 exerts on a Corpuscle placed at any Point n of the Superficies. — 5. We will 

 make, as above, bc = e, ce = e + e«j and also cn = e + ex, and half the 

 conjugate diameter of cn will be cg = e + ea — ex \ whence the radius of a 

 circle, equal in superficies to the ellipsis pg, will be a mean proportional be- 

 tween CB and CG, that is, e -j- ea — ^x. Therefore the spheroid beZ'^ exerts 

 the same attraction at n, as would be exerted at the pole of a spheroid ngof. 

 (fig. 11) of which the principal axis would be no = 2e -|- lex, and the second 

 would be to the principal, as I + « — f a to 1. 



Therefore in the expression of the attraction at the pole, (Art. 3) we must 

 substitute e -\- ex instead of e, and a — aa instead of a. But if _/ and g must 

 no longer be the same ; for we may easily perceive by the foregoing Theorem, 

 that the density must be the same in this spheroid ngof, at the distance r-\-rx 

 from the centre, as it is in the spheroid BE^e at the distance r. Therefore 

 f(-—^ -\- gir— y must be put instead oi fe -{■ ge . Thus we shall have 



3+p 3 + p X 5 + p 3 + p y. 5+ p 3 + q 3+qx5+q S + q X S+q 



for the attraction of the spheroid BEie at n. 



6. If we make x = a, the foregoing expression will be reduced to this 



2je^ ^cfe'+f» ^ej+9 2cge^ ^^.^^ expresses the attraction of the 



3 + p ' 5 + p ' 3 + q ' 5 + q ' ^ 



equator. 



7. If we would have the attraction at any point m within the spheroid, in 

 the expression of the attraction at n, we must put r instead of e. The proof „ 

 of this is plain from the same reasons that Sir Isaac Newton makes use of, | 

 (Corol. 3, Prop. Ql, I. 1, Princip. Math.) to show that the attraction of an '' 

 elliptic orb, at a point within it, is none at all. 



Problem IV. Let Rllrn- (fig. 12) be a Circle whose Centre my; it is re- 

 quired to find the Attraction which this Circle exerts on a Corpuscle at n, ac- 



