Page:Newton's Principia (1846).djvu/274

PROPOSITION X. PROBLEM III.

Suppose the uniform force of gravity to tend directly to the plane of the horizon, and the resistance to be as the density of the medium and the square of the velocity conjunctly: it is proposed to find the density of the medium in each place, which shall make the body move in any given curve line; the velocity of the body and the resistance of the medium in each place.

Let PQ, be a plane perpendicular to the plane of the scheme itself; PFHQ a curve line meeting that plane in the points P and Q; G, H, I, K four places of the body going on in this curve from F to Q; and GB, HC, ID, KE four parallel ordinates let fall from these points to the horizon, and standing on the horizontal line PQ, at the points B, C, D, E; and let the distances BC, CD, DE, of the ordinates be equal among themselves. From the points G and H let the right lines GL, HN, be drawn touching the curve in G and H, and meeting the ordinates CH, DI, produced upwards, in L and N: and complete the parallelogram HCDM. And the times in which the body describes the arcs GH, HI, will be in a subduplicate ratio of the altitudes LH, NI, which the bodies would describe in those times, by falling from the tangents; and the velocities will be as the lengths described GH, HI directly, and the times inversely. Let the times be expounded by T and t, and the velocities by $\scriptstyle {\frac {GH}{T}}$ and $\scriptstyle {\frac {HI}{t}}$; and the decrement of the velocity produced in the time t will be expounded by $\scriptstyle {\frac {GH}{T}}-{\frac {HI}{t}}$. This decrement arises from the resistance which retards the body, and from the gravity which accelerates it. Gravity, in a falling body, which in its fall describes the space NI, produces a velocity with which it would be able to describe twice that space in the same time, as Galileo has demonstrated; that is, the velocity $\scriptstyle {\frac {2NI}{t}}$: but if the body describes the arc HI, it augments that arc only by the length HI - HN or $\scriptstyle {\frac {MI\times NI}{HI}}$; and therefore generates only the velocity $\scriptstyle {\frac {2MI\times NI}{t\times HI}}$. Let this velocity be added to the beforementioned decrement, and we shall have the decrement of the velocity arising from the resistance alone, that is, $\scriptstyle {\frac {GH}{T}}-{\frac {HI}{t}}+{\frac {2MI\times NI}{t\times HI}}$.