Page:Newton's Principia (1846).djvu/279

&c. The second term $\scriptstyle {\frac {m}{n}}o-{\frac {bb}{aa}}o$ of this series is to be used for Qo; the third $\scriptstyle {\frac {bb}{a^{3}}}o^{2}$, with its sign changed for Ro²; and the fourth $\scriptstyle {\frac {bb}{a^{4}}}o^{3}$, with its sign changed also for So³, and their coefficients $\scriptstyle {\frac {m}{n}}-{\frac {bb}{aa}}$, $\scriptstyle {\frac {bb}{a^{3}}}$ and $\scriptstyle {\frac {bb}{a^{4}}}$are to be put for Q, R, and S in the former rule. Which being done, the density of the medium will come out as $\scriptstyle {\frac {\frac {bb}{a^{4}}}{{\frac {bb}{a^{3}}}{\sqrt {1+{\frac {mm}{nn}}-{\frac {2mbb}{naa}}+{\frac {b^{4}}{a^{4}}}}}}}$ or $\scriptstyle {\frac {1}{\sqrt {aa+{\frac {mm}{nn}}aa-{\frac {2mbb}{n}}+{\frac {b^{4}}{aa}}}}}$, that is, if in VZ you take VY equal to VG, as $\scriptstyle {\frac {1}{XY}}$. For aa and $\scriptstyle {\frac {m^{2}}{n^{2}}}a^{2}-{\frac {2mbb}{n}}+{\frac {b^{4}}{aa}}$ are the squares of XZ and ZY. But the ratio of the resistance to gravity is found to be that of 3XY to 2YG; and the velocity is that with which the body would describe a parabola, whose vertex is G, diameter DG, latus rectum $\scriptstyle {\frac {XY^{2}}{VG}}$. Suppose, therefore, that the densities of the medium in each of the places G are reciprocally as the distances XY, and that the resistance in any place G is to the gravity as 3XY to 2YG; and a body let go from the place A, with a due velocity, will describe that hyperbola AGK.   Q.E.I.

Example 4. Suppose, indefinitely, the line AGK to be an hyperbola described with the centre X, and the asymptotes MX, NX, so that, having constructed the rectangle XZDN, whose side ZD cuts the hyperbola in G and its asymptote in V, VG may be reciprocally as any power DNⁿ of the line ZX or DN, whose index is the number n: to find the density of the medium in which a projected body will describe this curve.

For BN, BD, NX, put A, O, C, respectively, and let VZ be to XZ or DN as d to e, and VG be equal to $\scriptstyle {\frac {bb}{DN^{n}}}$; then DN will be equal to A - O, $\scriptstyle VG={\frac {bb}{{\overline {A-O|}}^{n}}}$, $\scriptstyle VZ={\frac {d}{e}}{\overline {A-O}}$, and GD or NX - VZ - VG equal to $\scriptstyle C-{\frac {d}{e}}A+{\frac {d}{e}}O-{\frac {bb}{{\overline {A-O|}}^{n}}}$. Let the