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and consequently the velocity is inversely as the perpendicular let fall from the centre of force on a tangent to the curve at the body's place (L). (See Newton, Principia, ed. 3. p. 40.)

If the force or forces, whatever they be, do not act in the direction of a line drawn from the body's place to their centre, then the force T is not = 0, and the second equation will become d (p2.dv) ± p.Tdt2 = 0,

and, integrating,

p2 dv=h. dt = dtfp.Tdt;

consequently, by reason of the last term, the equable description of areas is, in this case, no longer preserved, or, in other words, is disturbed; and the force is called a disturbing force, because the centripetal force urging L towards T, is imagined to be the proper and natural force, by the action of which alone, the regular and equable description of areas would take place.

Hitherto no mention has been made of the law of the force. In the next Chapter, we will suppose L to be acted on solely by a centripetal force, and that force to vary, as it does in nature, according to the law of the inverse square of the distance between the

and the halves of the left-hand equations represent, respectively, the differentials of the areas on the planes of x, y, of x, z, and of y, z: or, are the projections of the incremental area (r2 dw) lying in the plane of the orbit; and, by the theory of projections,

h2 = c2 + c22+c".

This process has been inserted in a note, because it is not essential to the result which has been differently deduced in the text; and, it has been inserted partly on account of the importance and the celebrity of its result, and partly as a kind of exercise to the Student, and as a means of shewing how the same conclusions may be obtained either from the fundamental equations of p. 8, or the transformed ones of P. 10.

body or point attracted, and the centre of force or attraction. According to the first condition then, Kepler's Law must, in this case, accurately obtain. And the second condition will conduct us to results equally curious with those that have been already obtained, and to the establishment of two other of Kepler's Laws relative to the form of the orbit and the variation of the periodic time.

CHAP. III.

The Centripetal Force is supposed to act inversely as the Square of the Distance. Consequences that flow from it. The Orbit, or the Curve described by the moving Body round the Central, an Ellipse. Kepler's Law of the Squares of the Periodic Times varying as the Cubes of the Major Axes. Kepler's Problem for determining the true from the mean Anomaly. His Law respecting the Periodic Times not exactly

true.

THE centripetal force tending towards the point or centre T being, by supposition, the sole force that acts on the body, the perpendicular force, which has been designated by T, must, for the reasons already assigned in p. 14, be equal nothing; and, since the body can never deviate from that plane in which it once has moved, we may get rid of, or expunge from the calculation, the force S, by supposing the plane, to which its action is perpendicular, coincident with the plane of the orbit. If, besides these conditions, we assume to be an invariable quantity, and expound the centripetal force P by, (which is to suppose the law of its variation to be according to the inverse square of the distance), the equations (4), (5), of p. 10, will become

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2 du

w.dr + rd2 w = 0,

p becoming in this case r, and v, w.

Now, as we have seen, the second of these equations gives us Kepler's law of the equable description of areas, and the variation of the velocity in terms of the perpendicular. The first, if dt were eliminated, would giver in terms of w and certain constant quantities: it would then give us, what is the object of enquiry, namely, the nature of the curve described. The end to be attained

then is very obvious. By means of the two equations in which d (see p. 14,) is constant, we must form another from which dt has been eliminated, and containing dw as a constant element.

The conversion of one equation, in which dt is constant and dw variable, into another in which dw should be constant and dt variable, is (see Dealtry's Fluxions, p. 328. Vince, p. 185. ed. 1. Prin. Anal. Cale p. 90,) a common analytical operation: and so simple, that it may be here inserted without its materially impeding the progress of investigation.

The first equation, employing the general character P instead of, and (since there is, in this case, no necessity for distinction) vinstead of w, becomes

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and, the second equation integrated (see p. 14,) gives

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and the differential of this, supposing (see 1. 5,) dt variable, and

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now the quantity, within the brackets, is equal d (~~);

dr

; there

fore, if we make u =

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and consequently, -'du==+, there

results

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which is an equation between u and v, and which integrated

), and

would give us the relation between u ( = ;),

v, and con

sequently would determine (see p. 21,) the nature of the curve described. Our attention is therefore naturally directed to the integration of the preceding equation.

If, in the equation += 0, we substitute, instead of u,

d2 u dv2

either a. sin. v, or b. cos. v, the resulting equation becomes, as it is technically said, identically nothing. Hence, either ua sin. v, or u = b cos. v, satisfies the differential equation; so must u = a sin. v+b cos v, and since (see Prin Anal. Calc. pp. 90, &c.) an equation of the second degree requires for its complete integration two arbitrary quantities, the last form, viz. u=a sin. v+b. cos. v, must be the true and complete one: and the condition, either that a = 0, or b = 0, can only happen in particular cases.

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bitrary quantities a and b are to be determined by the conditions.

> h'

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= a. cos. V b. sin. v; when v = 0, the value

d v

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quently, between the values of v = 0, and v = 90°, there is a

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