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

280 which the given particle in the place Ff would attract the ame corpucle.

For if we conider firt the force of the phærical uperficies 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 uperficies generated by the revolution of the arc rE will be as the lineola Dd, the radius of the phere PE remaining the ame; as Archimedes has demontrated in his book of the phere and cylinder. And the force of this uperficies exerted in the direction of the lines PE or Pr ituate all round in the conical uperficies, will be as this annular uperficies it elf; that is as the lineola Dd, or which is the ame as the rectangle under the given radius PE of the phere and the lineola Dd; but that force, exerted in the direction of the line PS tending to the centre S, will be les in the ratio of PD to PE, and therefore will be as FD x Dd. Suppoe now the line DF to be divided into innumerable little equal particles, each of which call Dd; and then the uperficies FE will be divided into o many equal annuli, whoe forces will be as the um of all the rectangles PD x 'Dd, that is, as $$\scriptstyle \frac 12PF^2 - \frac 12PD^2$$, and therefore as $$\scriptstyle DE^2$$. Let now the uperficies FE be drawn into the altitude Ff; and the force of the olid EFfe exerted upon the corpucle 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 corpucle P at the diŧance PF. But if that forte be not given, the force of the olid EFfe will be as the olid $$\scriptstyle DE^2 \times Ff$$ and that force not given, conjunctly. Q. E. D.