Page:Newton's Principia (1846).djvu/281

 to $$\scriptstyle \left(\frac{HT}{AC}\right)^{n-1}$$, or of $$\scriptstyle \frac{S^2}{R^{4-n}}$$ to $$\scriptstyle \left(1+QQ\right)^{n-1}$$ may be given; the body, in an uniform medium, whose resistance is as the power Vn of the velocity V, will move in this curve. But let us return to more simple curves.

Because there can be no motion in a parabola except in a non-resisting medium, but in the hyperbolas here described it is produced by a perpetual resistance; it is evident that the line which a projectile describes in an uniformly resisting medium approaches nearer to these hyperbolas than to a parabola. That line is certainly of the hyperbolic kind, but about the vertex it is more distant from the asymptotes, and in the parts remote from the vertex draws nearer to them than these hyperbolas here described. The difference, however, is not so great between the one and the other but that these latter may be commodiously enough used in practice instead of the former. And perhaps these may prove more useful than an hyperbola that is more accurate, and at the same time more compounded. They may be made use of, then, in this manner.

Complete the parallelogram XYGT, and the right line GT will touch the hyperbola in G, and therefore the density of the medium in G is reciprocally as the tangent GT, and the velocity there as $$\scriptstyle \sqrt{\frac{GT^{2}}{GV}}$$; and the resistance is to the force of gravity as GT to $$\scriptstyle \frac{2nn+2n}{n+2}\times GV$$.

Therefore if a body projected from the place A, in the direction of the right line AH, describes the hyperbola AGK and AH produced meets the asymptote NX in H, and AI drawn parallel to it meets the other asymptote MX in I; the density of the medium in A will be reciprocally as AH, and the velocity of the body as $$\scriptstyle \sqrt{\frac{AH^{2}}{AI}}$$, and the resistance there to the force of gravity as AH to $$\scriptstyle \frac{2nn+2n}{n+2}\times AI$$. Hence the following rules are deduced.

1. If the density of the medium at A, and the velocity with which the body is projected remain the same, and the angle NAH be changed, the lengths AH, AI, HX will remain. Therefore if those lengths, in any