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QR + rn - pt,

and the tangential force, or, if the orbit be circular, the force perpendicular to TL, is Lt*: and such, although somewhat differently deduced and stated, are Newton's resolutions of force in his eleventh Section.

To like resolutions, or, as they may be called, Modifications of the Sun's disturbing force, Newton assigned, as to their special causes, certain of the Lunar inequalities. Of these the most notable is the Variation, as being the deviation from the most simple of Kepler's Laws, namely, the equable description of areas. Newton attributed this inequality, principally, to the tangential force (Lt). But, as we have seen, such a force would subsist, and would accelerate and retard the description of areas, or (stating the infringement of Kepler's Law in other terms) would make the planet now before and at another time behind its mean or its elliptical + place, if the force from which it is derived,

* It is easy to express these resolutions differently: thus,

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the whole centripetal force, therefore, see l. 1. of the text, is

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+ Elliptical, or found in the ellipse by correcting the mean place, on account of the first, or elliptical inequality, and by means of Kepler's Problem. See Astronomy, Chap. XVIII.

namely, that by which S draws L and T should not vary according to the inverse square of the distance. In order, then, to ascertain whether the latter law of the variation of the force, or any other law, be the true one, a closer examination of results must be instituted; in fact, the inequality itself must be computed, and in his third Book the Variation itself (see Prop. XXVI, XXIX.) is computed from those expressions of the force which are given in the Note of the preceding page, and which suppose the law of the force to be that of the inverse square of the distance. The agreement of the computed and observed variation proves the law of the force to have been rightly assumed, and, as far as a single instance can go, proves the truth of that law.

It is so with other popular explanations and strict computations of other Lunar inequalities: the former render probable the Theory of Gravitation, the latter, if their results agree with those of observation, confirm it.

The diagram which we have already introduced may serve to explain after what manner another Lunar inequality, may, on Newton's principles, be assigned, (as to its special cause), to a modification of a resolved part of the disturbing force.

The part of the disturbing force which acts in the direction of the radius vector LT is (see p. xxxvi.)

rn - pt,

which sometimes augments, at other times diminishes the centripetal force RQ that arises from the central body at T: or which, since we are speaking of the Lunar Theory, so affects the Moon's gravity to the Earth. The latter, the diminishing effect, predominates so that, during one synodic revolution, there takes place what may be called, a Mean Diminution of the Moon's Gravity. Hence ensues an augmentation of the Moon's periodic time beyond what would have been its value, had there been no disturbing force; and, the greater the mean diminution of gravity, the greater the augmentation of period. Now the diminution we are speaking of, is the result, or mean, of the several diminutions

and augmentations that happen during one synodic period, the Earth's distance being supposed to remain the same. If the distance be changed, the result, or mean, will be changed *.

* The computation is easily given.

Let m be the mass of the Sun

the radius of his orbit,

r the radius of the Lunar orbit,

y= SL, and 0 ▲ LTS, then, see p. xxxvii.

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Now the sum of the disturbing forces in the direction of the radius is the sum of the several parts on the right-hand side of the equation taken for every point in the circle, from 0° to 360°. The sum of all the

mr

21'3

is

- mr

x circumference. With regard to the second part, 2'3 since cos. 2(90°+0)=—cos. 20, whatever be the value of 0, there must be 3 m r as many terms of the form 2/3

cos. 28, as of the form

3 mr

23

cos. 20,

the corresponding terms being respectively distant from each other by

3 m r

90°. The sum, therefore, of all the terms

273 Cos. 20, 0 being taken of every value from 0 to 360°, must be nothing, the corresponding positive and negative terms destroying each other. The sum, therefore, of all

the rnpt is

will be

-mr
2 r'3

x circumference: consequently the mean effect

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which may be supposed constant during one synodic revolution: but which will vary when the Sun's distance from the Earth varies.

Every month, therefore, it will be changed. It will be increased by the Earth's approach to the Sun, and, consequently, will be greater in Winter than in Summer. Now these results which flow immediately from that modification of the resolved part of the disturbing force which acts in the direction of the radius, resemble the properties of the annual equation (see Astron. pp. 328, &c.): and, therefore, to go no farther, that irregularity in the Moon's motion, is also probably caused by the Sun's disturbing force and if exactly accounted for, that is, by the agreement of its computed with its observed quantity, would additionally confirm the Theory of Gravity *.

:

It was thus that Newton shewed that the Sun's disturbing force (a necessary consequence of the Theory of Gravity), if not their real cause, would at least account for two of the Lunar inequalities: the Variation discovered by Tycho Brahé, and the Annual Equation discovered by Kepler from the former Astronomer's observations.

But there existed, of much older date, and discovered by Hipparchus and Ptolemy, another considerable Lunar inequality called the Evection: and this, also, Newton shewed to be explicable by the theory of Gravity.

In the explanation of this inequality the eccentricity of the Moon's orbit is an essential condition; which is not the case with the explanation of the two former inequalities. The Evection arises from the variation of the eccentricity of the orbit: and, the variation of the eccentricity from a modification of the general effect of a resolved part of the disturbing force dependent on the position of the apsides.

* Hisce motuum Lunarium computationibus ostendere volui, quod motus Lunares, per Theoriam Gravitatis, a causis suis computari possint. Per eandem Theoriam inveni præterea quod Æquatio annua medii motus Lunæ oriatur a variâ dilatione orbis Lunæ per vim Solis, juxta, Cor. 6. Prop. LXVI. Lib. 1. Scholium, Lib. 14,

The general effect of that part of the disturbing force which acts in the direction of the radius is to augment the Moon's Gravity in quadratures, and to diminish it in syzygies. Its peculiar effect when the apsides lie in quadratures (which is a modification of its general effect) is, since it varies as the distance, to augment the Perigean Gravity, which is the greatest, by the least quantity, and to augment the Apogean Gravity, which is the least, by the greatest quantity. Its peculiar effect, then, in this position of the apsides, is to make the ratio bet ween the Perigean and Apogean Gravities less than that of the inverse square of the distances, and thereby to render the orbit less eccentric. When the apsides lie in syzygies, the peculiar effect of the resolved part of the disturbing force is to render the orbit more eccentric, by rendering the ratio between the Perigean and Apogean Gravities greater than that of the inverse square of the least and greatest distances; because, in this position of the axis major, the Perigean Gravity is diminished by the least and the Apogean by the greatest quantity. Now the eccentricity being, as it is in the first position of the apsides, diminished, the Equation of the Centre (see Astron. p. 202.) will be necessarily diminished; and in the second position the eccentricity and the Equation of the Centre will be increased. In intermediate positions the effect will be a blended one.

But, whatever the position, the variations of the Equation of the centre as consequences deduced from the disturbing force, correspond to the observed and registered properties of the Evection. The former, therefore, will account for the latter. The inequality is an effect of which that modification of the Sun's disturbing force just spoken of seems to be the adequate cause (see Astron. pp. 325, &c.).

It was nearly after the preceding manner that Newton, in his eleventh Section, explained the three principal Lunar inequalities, by his Theory of Gravity; and, by such explanations, added new proofs of the truth of that theory: the proofs being, as it has been already mentioned, of a different kind and of a higher scale than those which had been derived from the establishment of two of Kepler's Laws.

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