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body's orbit. We may suppose, then, the latter plane and that in which TI, T are to be coincident; in which supposition, s and S=0, and p = r, and T, also, = 0; for, the sole force (P) acting in the direction LT admits of no resolution in a direction perpendicular to LT. The equation (6), then, of p. 10, disappears, and the equations (4), (5), are reduced to these forms, d2r - r.dv2 + Pdt2 = 0,

2 dvdr + rd2 v = 0.

The second of these equations may be put under this form, 2rdr.dv + r2d2v, or, d(r2d v) = 0.

But, if d (r2dv) = 0, we have, by integration, r2dv = hdt,

dt being the differential of the time, which is supposed constant, and h being an arbitrary quantity to be determined according to the conditions of any specific case.

The result just obtained is a very remarkable one; it amounts to what is usually known by the name of Kepler's Law of the Equable Description of Areas. For, if the body be describing the A

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curve Lpq, and Tp be supposed to be indefinitely near to 1L, (pn being perpendicular to LT), the incremental area

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and, hence, the differential of the area will be proportional to h.dt and dt, and, accordingly, the integral or the whole area (LTq for instance), will be proportional to the time of the body's describing the arc L q.

This law of the equable description of areas, Kepler, by observation, ascertained to be true in the orbits of the planets; and Newton, in the first Proposition of the second Section of the Principia, shewed that it was a necessary consequence of the action of a centripetal force on a body moving obliquely to a line joining it and the centre of force.

The law, indeed, depends on the condition of the force, or forces, being centripetal : that is, it requires they should act in a line joining the body, and the point which is considered as the centre of the body's motion. In other respects there is no limitation; the force may be of any quantity, and may vary according to any law.

Since the body, in the case we have supposed, can never be solicited to leave the plane in which it first moves, we have, for the sake of simplicity, considered only that plane; and, solely for that reason. For, if we assume a plane in which lies and inclined to that of the orbit, we may obtain results the same as the preceding.

In this case, the second equation is

2 dv.dpp. d2 v = 0,

or d (p2. dv) = 0,

in which du is, as in p. 10, the incremental angle contained between two contiguous radii, p and p + dp. Now this equation integrated gives, like the former,

p2.dv =h. dt.

But pdv, in this case, is the projection of the differential of the area (r2. dw, dw being the incremental angle or the dif

* Astronomy, p. 188.

ferential angle between two contiguous radii r and r+dr); which also is constant when dt is; consequently, the area of the projected curve varies as the time: and from this result the former (see p. 14, l. 12,) might have been deduced; since, by the principles of projection, the area really described (fr2 dw), is to the projection of the area (p2 dv) in the ratio of radius to the cosine of the inclination of the two planes: which, since the inclination of the plane of the orbit can never alter, must be a constant ratio.

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The preceding result may, with little difficulty, be more formally deduced; thus, if denote the inclination of the planes

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LAT, tAT*, that is, if it equal (see Trig. ed. 2. p. 119.) the spherical angle LAt, we have, by Naper's Rules, (see Trig. p. 136.) 1 x cos. LAt = tan. At x cot, LA, or,

tan. v=tan. w x cos. ;

* In the Figure of the text, the dotted line Al represents the curve AL projected on the plane of Al, IK: and At represents part of a great circle lying in the last plane.

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Hence, the relation between the quantities h and h' is thus defined,

h' = h. cos. p*.

* Kepler's Law of the equable description of areas has been proved from the equations (4), (5), (6) ; but, it may readily be proved from the original equations [1], [2], [3]: thus

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multiply [2] by x, and subtract from it [1] multiplied by y; multiply [3] by x, and subtract from [1] multiplied by z, &c. then the following equations will arise

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Now, the sole force (F), being that which acts in the direction of

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If we call the velocity with which Lp is described, we

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but Lp x TM - P x LT-dh.dt.

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=

=

2

2

=

2

consequently, Xy - Yx, Xz-Zx, Yz-Zy, are all = 0;

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Hence, integrating and assuming c, c', c", three arbitrary quantities, there result

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