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which belongs to the problem of two bodies and the elliptical theory; and its integration gives us

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the known equation, (see Vince's Conic Sections) of an ellipse.

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If this value of u be substituted, as an approximate value of u, in the differential equation constructed by taking in some of the larger terms of the disturbing force, (supposed to vary as the distance and to act solely in the direction of the radius vector) there results, when the method is corrected, (see p. 111, &c.) an equation of this form

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which, however, (see p. 110.), since some terms, by reason of the small eccentricity of the orbit, are neglected, is only an approximate value.

This last solution, considered as an analytical one, is similar to the former,

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which is an equation to an ellipse, v being the angle contained between the axis major and the radius vector; but

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is not an equation to an ellipse when v is the anomaly. Still, which is curious, the body's place, as determined by the preceding equation, can be found by means of a construction, of which an ellipse is the essential part. Thus,

Let be the apside, C the centre of motion, and let the angle pv: then, if

1 Cp

é

(u) be assumed = +

cos. cv, p

1
L L

is the body's place, and Vp described by p, (or the locus of the

extremity of the line determined by the preceding equation) is part

K

of the body's orbit. Thus far is independent of an ellipse and of every other curve; we now come to the construction.

=

=

C being the focus, CV part of the axis major, describe an ellipse the semi-parameter of which shall L, and the ratio of its eccentricity to the semi-axis shall e. Incline the line Cu(CV) so to CN, that the angle VCu shall equal v cv, and on Cu describe an ellipse similar to that which has been already described on CV, then, since

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uCpVCpVCu v (v - cv) = cv,

1

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is the equation to the ellipse up K, and, accordingly, p the body's place is in such ellipse.

By means of this device then and construction, the body may be supposed to be always moving in a moveable ellipse. And it was under this point of view that Newton, when he made the first modification of, or departure from, the strictly elliptical theory, considered the planetary motions, (see Princ. Sect. IX.),

The preceding construction does not obtain, except (see &c. are rejected: it is

pp. 110, &c.) such terms as ( − «)*,

P

only true, therefore, in orbits of very small eccentricity *. There

* Circulis maximi finitimi. Princ. Sect. IX. Prop. xiv.

is one variation of the disturbing force, however, in which the construction will hold, whatever be the eccentricity of that orbit: if that force varies inversely as the cube of the distance, or, if P=

pu2 + Ku3, then (see pp. 109, &c.) N =

equation is exactly

d'u

dv2

Ku

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and the differential

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and this corresponds to what Newton has proved in the fortyfourth Proposition of the ninth Section.

The analytical formulæ that have been deduced, when translated into the language of curves, correspond exactly to the results obtained by Newton in his ninth Section: but they are deduced from cases entirely fictitious. The disturbing forces which act in Nature do not act solely in the direction of the radius; and, since this is Newton's supposition in the abovementioned Section, the Propositions therein contained cannot explain completely the Planetary Phenomena.

One of the most noted of those phenomena, is the progression of the Lunar Apogee; and, probably with a view to its explanation, Newton originally constructed the ninth Section: a section, more than any other, abounding with curious, novel, and refined methods.

It is true, that a disturbing force acting solely in the direction of the radius, such as has been supposed in the preceding instances, will cause a progression of the apogee; and, it is evident, would, by assuming the Sun's disturbing force of a convenient magnitude, give, as a result of calculation, the just value of that progression. This, however, is not to explain the phenomenon ; since, in order to obtain the above-mentioned value, it is necessary to assume the Sun's disturbing force nearly the double of what it really is.

The chief merit, then, of that ingenious section (the 9th) of

the Principia, consists in the idea of a moveable ellipse. To this (we may conjecture) Newton was led, by Kepler's discoveries and his own investigations, which established the nearly elliptical forms of the orbits of the planets, and by the results of Astronomical observations which shewed the Aphelia of those orbits to be progressive.

There have been mathematicians, however, who have wished to discover in that section more than Newton meant it should contain, and have dispensed with the tangential disturbing force, although its operation is as certain as that of the disturbing force which acts in the direction of the radius. And this is strange, since there are no probable or paramount arguments, by which it can be made to appear that, in the investigation of the progression of the Lunar Apogee, a right result is to be looked for, when one source of that inequality is rescinded.

Newton, it is true, no where affirms that the progession cannot be determined by the principles, and according to the method of the ninth Section; nor, as it is known, has he given a solution of that problem. He says, Scholium, Prop. xxxv. ed. 1. that he had found by calculation, the quantity of the progression; but, the method either did not completely satisfy him, or did not harmonize with the stile of his other investigations.

The question of the progression of the Lunar Apogee, and the analytical method of determining its quantity, will be resumed in another part of this Treatise. We must now regain the direct course of investigation; and, as it has been already suggested, the next attempts ought to be directed towards the conversion of II into a series of cosines, such as A cos. mv +

The relative beauty and accuracy of the geometrical and analytical methods is a point not easily decided on. But, their relative power and efficiency may be estimated. Physical Astronomy presents to us various cases, in which the analytical method has succeeded in affording true results, whilst the geometrical has failed. The one in the text, the progression of the Lunar Apogee, has never been determined by the latter method.

B cos. nv + &c. Instead, however, of attempting that on a general scale, we prefer (with a view to the interests of the Student) to proceed by instances: beginning with the most simple, and passing on to others that become more complex by the largeness of the disturbing force, and by the obliquity of the direction of its action to that of the centripetal force.

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