Page:Newton's Principia (1846).djvu/296

the time of the same revolutions will be as $\scriptstyle {\frac {OP}{OS}}$, that is, as the secant of the same angle, or reciprocally as the density of the medium.

Cor. 7. If a body, in a medium whose density is reciprocally as the distances of places from the centre, revolves in any curve AEB about that centre, and cuts the first radius AS in the same angle in B as it did before in A, and that with a velocity that shall be to its first velocity in A reciprocally in a subduplicate ratio of the distances from the centre (that is, as AS to a mean proportional between AS and BS) that body will continue to describe innumerable similar revolutions BFC, CGD, &c., and by its intersections will distinguish the radius AS into parts AS, BS, CS, DS, &c., that are continually proportional. But the times of the revolutions will be as the perimeters of the orbits AEB, BFC, CGD, &c., directly, and the velocities at the beginnings A, B, C of those orbits inversely; that is as $\scriptstyle AS^{\frac {3}{2}}$, $\scriptstyle BS^{\frac {3}{2}}$, $\scriptstyle CS^{\frac {3}{2}}$. And the whole time in which the body will arrive at the centre, will be to the time of the first revolution as the sum of all the continued proportionals $\scriptstyle AS^{\frac {3}{2}}$, $\scriptstyle BS^{\frac {3}{2}}$, $\scriptstyle CS^{\frac {3}{2}}$, going on ad infinitum, to the first term $\scriptstyle AS^{\frac {3}{2}}$; that is, as the first term $\scriptstyle AS^{\frac {3}{2}}$ to the difference of the two first $\scriptstyle AS^{\frac {3}{2}}-BS^{\frac {3}{2}}$, or as ⅔AS to AB very nearly. Whence the whole time may be easily found.

Cor. 8. From hence also may be deduced, near enough, the motions of bodies in mediums whose density is either uniform, or observes any other assigned law. From the centre S, with intervals SA, SB, SC, &c., continually proportional, describe as many circles; and suppose the time of the revolutions between the perimeters of any two of those circles, in the medium whereof we treated, to be to the time of the revolutions between the same in the medium proposed as the mean density of the proposed medium between those circles to the mean density of the medium whereof we treated, between the same circles, nearly: and that the secant of the angle in which the spiral above determined, in the medium whereof we treated, cuts the radius AS, is in the same ratio to the secant of the angle in which the new spiral, in the proposed medium, cuts the same radius: and also that the number of all the revolutions between the same two circles is nearly as the tangents of those angles. If this be done every where between every two circles, the motion will be continued through all the circles. And by this means one may without difficulty conceive at what rate and in what time bodies ought to revolve in any regular medium.