EXAMPLE 2. Let the line PFQ be a parabola, having its axis AF perpendicular[Pg 272] to the horizon PQ, to find the density of the medium, which will make a projectile move in that line. From the nature of the parabola, the rectangle PDQ is equal to the rectangle under the ordinate DI and some given right line; that is, if that right line be called b; PC, a; PQ, c; CH, e; and CD, o; the rectangle a + o into c - a - o or ac - aa - 2ao + co - oo, is equal to the rectangle b into DI, and therefore DI is equal to . Now the second term of this series is to be put for Qo, and the third term for Roo. But since there are no more terms, the co-efficient S of the fourth term will vanish; and therefore the quantity , to which the density of the medium is proportional, will be nothing. Therefore, where the medium is of no density, the projectile will move in a parabola; as Galileo hath heretofore demonstrated. Q.E.I. EXAMPLE 3. Let the line AGK be an hyperbola, having its asymptote NX perpendicular to the horizontal plane AK, to find the density of the medium that will make a projectile move in that line. Let MX be the other asymptote, meeting the ordinate DG produced in V; and from the nature of the hyperbola, the rectangle of XV into VG will be given. There is also given the ratio of DN to VX, and therefore the rectangle of DN into VG is given. Let that be bb: and, completing the parallelogram DNXZ, let BN be called a; BD, o; NX, c; and let the given ratio of VZ to ZX or DN be . Then DN will be equal to a - o, VG equal to , VZ equal to , and GD or NX - VZ - VG equal to . Let the term be resolved into the converging series , &c., and GD will become equal to &c. The second term [Pg 273] of this series is to be used for Qo; the third , with its sign changed for Ro2; and the fourth , with its sign changed also for So3, and their coefficients , and 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 , or , that is, if in VZ you take VY equal to VG, as . For aa and are the squares of XZand 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 . 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.