Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/366

which the given particle in the place Ff would attract the ſame corpuſcle.

For if we conſider firſt the force of the ſphærical ſuperficies FE which is generated by the revolution of the arc FE, and is cut any where, as in r, by the line de; the annular part of the ſuperficies generated by the revolution of the arc rE will be as the lineola Dd, the radius of the ſphere PE remaining the ſame; as Archimedes has demonſtrated in his book of the ſphere and cylinder. And the force of this ſuperficies exerted in the direction of the lines PE or Pr ſituate all round in the conical ſuperficies, will be as this annular ſuperficies it ſelf; that is as the lineola Dd, or which is the ſame as the rectangle under the given radius PE of the ſphere and the lineola Dd; but that force, exerted in the direction of the line PS tending to the centre S, will be leſs in the ratio of PD to PE, and therefore will be as FD x Dd. Suppoſe now the line DF to be divided into innumerable little equal particles, each of which call Dd; and then the ſuperficies FE will be divided into ſo many equal annuli, whoſe forces will be as the ſum of all the rectangles PD x 'Dd, that is, as $\scriptstyle {\frac {1}{2}}PF^{2}-{\frac {1}{2}}PD^{2}$, and therefore as $\scriptstyle DE^{2}$. Let now the ſuperficies FE be drawn into the altitude Ff; and the force of the ſolid EFfe exerted upon the corpuſcle P will be as $\scriptstyle DE^{2}\times Ff$; that is, if the force be given which any given particle as Ff exerts upon the corpuſcle P at the diſŧance PF. But if that forte be not given, the force of the ſolid EFfe will be as the ſolid $\scriptstyle DE^{2}\times Ff$ and that force not given, conjunctly. Q. E. D.