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 ; then DN will be equal to A - O, , , and GD or NX - VZ - VG equal to . [Pg 274]Let the term be resolved into an infinite series , &c., and GD will be equal to , &c. The second term of this series is to be used for Qo, the third for Roo, the fourth for . And thence the density of the medium , in any place G, will be and therefore if in VZ you take VY equal to n × VG, that density is reciprocally as XY. For A2 and are the squares of XZ and ZY. But the resistance in the same place G is to the force of gravity as to 4RR, that is, as XY to . 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 or . Q.E.I. SCHOLIUM. In the same manner that the density of the medium comes out to be as , 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 . And therefore if a curve can be found, such that the ratio [Pg 275]of to or of to 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 ; and the resistance is to the force of gravity as GT to . 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 , and the resistance there to the force of gravity as AH to . Hence the following rules are deduced.