TOL. XL.] PHILOSOPHIC AI. TSANSACTIONS. ill 



the rotation of NnmM will be -(2ru — uu)du, where c is the circumference 



to the radius r. 



Now because of the smallness of nm, we may account all the particles of 

 matter contained in that space as equally attracting the corpuscle at a ; there- 

 fore we shall have the attraction of that small space, if we multiply that solidity 



• 1 A P 



by the attraction at m, and that attraction at m is — ; X — . Thus will be had 



-' ' AM* AM 



analytically ^^"^^ .j,2ru—uu.du = j^r^Tsr^^''''" ^ " ~ udu^u) the in- 

 tegral of which, - — — (4-r«v^« — 4-w"v/w)> is the attraction of the space 

 arising from the rotation of anm. In which value if we take u = 2r, we have 

 by reduction -^ac. Hence the attraction of the whole space aeqc is expressed ; 

 then adding ^ for the attraction of the whole sphere, we have -j-c -f- -^c for 

 the attraction of the ellipsoid. 



Carol. — For an oblong spheroid a will be negative, and the united attraction 

 will be -I* — tVc. 



Note. — If the foregoing spheroid, instead of circular elements arising in pn, 

 consist of other elements, for instance elliptical, which should differ from a ' 

 circle no more than the ellipsis ae, and by which there would be the same super- 

 ficies as by the circles pn, it appears that the attraction will be still the same, 

 because in those elements pn, whatever the remaining force might be, the circles 

 PM being taken away, it will be as it were composed of parts which would have 

 the same attraction as on that of the ellipsoid, having regard to the smallness 

 of NM, and to the equable quantity of matter. 



Lemma. Let kl be a circle, fig. 1 3, h its centre, vh perpendicular to the 

 area of the circle, and nh = vh, but making with it an angle infinitely small, 

 or very small: then the attraction of the circle kl on n, may be taken, without 

 sensible error, as the attraction of the circle on v; or, which is the same thing, 

 that the one attraction differs from the other only by a quantity infinitely less, 

 with respect to both, than as vn is less in respect to hv. 



To demonstrate which proposition, it must be shown, that two corpuscles 

 being placed at the extremity of any diameter kl, there is one attractive force 

 at N, and another force at v, the sum of which may be accounted the same. 

 But, neglecting the computation for having the attraction of the body at k on 

 the corpuscle at n, it will be easily seen that it will be the same with the attrac- 

 tion on V, to which should be added a small quantity involving nv. In like 

 manner it may be seen, that the attraction of a body at l on the corpuscle n, 

 will be the same as the attraction on v, deducting the same small quantity. 

 Therefore the sum of both these attractions is one and the same. 



VOL. VIII. R 



