Page:Newton's Principia (1846).djvu/280

term $\scriptstyle {\frac {bb}{{\overline {A-O|}}^{n}}}$ be resolved into an infinite series $\scriptstyle {\frac {bb}{A^{n}}}+{\frac {nbb}{A^{n+1}}}\times O+{\frac {nn+n}{2A^{n+2}}}\times bb\ O^{2}+{\frac {n^{3}+3nn+2n}{6A^{n+3}}}\times bb\ O^{3}$,&c.,and GD will be equal to $\scriptstyle C-{\frac {d}{e}}A+{\frac {bb}{A^{n}}}+{\frac {d}{e}}\ O-{\frac {nbb}{A^{n+1}}}\ O-{\frac {+nn+n}{2A^{n+2}}}bb\ O^{2}-{\frac {+n^{3}+3nn+2n}{6A^{n+3}}}bbO^{3}$, &c. The second term $\scriptstyle {\frac {d}{e}}\ O-{\frac {nbb}{A^{n+1}}}\ O$ of this series is to be used for Qo, the third $\scriptstyle {\frac {nn+n}{2A^{n+2}}}bb\ O^{2}$ for Roo, the fourth $\scriptstyle {\frac {n^{3}+3nn+2n}{6A^{n+3}}}bbO^{3}$ for So³. And thence the density of the medium $\scriptstyle {\frac {S}{R{\sqrt {1+QQ}}}}$, in any place G, will be $\scriptstyle {\frac {n+2}{3{\sqrt {A^{2}+{\frac {dd}{ee}}A^{2}-{\frac {2dnbb}{eA^{n}}}A+{\frac {nnb^{4}}{A^{2n}}}}}}}$, and therefore if in VZ you take VY equal to n $\scriptstyle \times$ VG, that density is reciprocally as XY. For A² and $\scriptstyle {\frac {dd}{ee}}A^{2}-{\frac {2dnbb}{eA^{n}}}A+{\frac {nnb^{4}}{A^{2n}}}$ are the squares of XZ and ZY. But the resistance in the same place G is to the force of gravity as 3S $\scriptstyle \times {\frac {XY}{A}}$ to 4RR, that is, as XY to $\scriptstyle {\frac {2nn+2n}{n+2}}$ VG. And the velocity there is the same wherewith the projected body would move in a parabola, whose vertex is G, diameter GD, and latus rectum $\scriptstyle {\frac {1+QQ}{R}}$ or $\scriptstyle {\frac {2XY^{2}}{nn+n\times VG}}$.   Q.E.I.

SCHOLIUM.

In the same manner that the density of the medium comes out to be as $\scriptstyle {\frac {S\times AC}{R\times HT}}$, in Cor. 1, if the resistance is put as any power Vⁿ of the velocity V, the density of the medium will come out to be as $\scriptstyle {\frac {S}{R^{\frac {4-n}{2}}}}\times \left({\frac {AC}{HT}}\right)^{n-1}$

And therefore if a curve can be found, such that the ratio of $\scriptstyle {\frac {S}{R^{\frac {4-n}{2}}}}$ to