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of which terms it will be sufficient, on the score of exactness, to retain the first.

The differential equation, if we now collect its several terms,

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(g+ 1) (1 − m) e y sin. (2 v − 2 mv-gv+cv + 0)

(g−1)(1 + m) e y sin. (2 v − 2 m v+gv-cv-0)

·(g + 1) (1 +m) e y sin. (2v – 2mv – g v− c v−0)

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(g − 1) (1 — m) e y sin. (2v − 2 mv+go+cv− 0)

é'y sin. (gv + cm v — 0)

é'y sin. (gv - dmv - 0)

a_(g+1) e'y sin. (2 v − 2 m v¬go + c'mv+ 0)

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(g+1) é' y . sin. (2 v−2 m v−g v − ć′ mv 4 0)

II

This equation is similar to the equation of p. 156, and admits of a similar integration; of such, indeed, as was explained in pp. 100, &c. The integration gives the value of s, and, in the case before us, will express it by a series of terms of which the arguments are the same as in the preceding differential equation, namely, gv 0, 20 2 mv - gv 0, g v + c v − 8, &c. &c., and the coefficients, the corresponding coefficients in the differential equation divided respectively by

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g2 — 1, (2 − 2 m − g)2 − 1, (g + c)2 − 1, &c.

(see p. 100.)

The terms representing the value of s mathematically expound, as in the preceding cases of the values of u and v, certain equations that serve to correct the latitude. Under a merely mathematical view, the terms, and, consequently, the equations, are infinite in number. But it is sufficient to retain a few: those that are, on account of their magnitude, eminent above the rest. And, as in the former cases, when the parallax and longitude were determined, so in this it may be shewn why some terms are much larger than others. The two first terms, for instance, that by integrating the preceding differential equation, will express s,

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3 Ka
a. (g2-1)

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(2—2m-g)3− 1 ( 1+g+1-82) y sin. (2v−2mv−gv+0).

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Now each of these terms is large, by reason of the smallness of its denominator. The first expounds the Equation, the argument of which is termed the Argument of Latitude: the second expounds the principal equation of latitude, and which, from mere observation alone, was discovered by Tycho Brahe.

The coefficient of the first term may be used for determining the Regression of the Lunar Nodes, just as the coefficient of cos. cv in the value of u, was used by Clairaut for determining the Progression of the Apogee: thus the value of s in the undisturbed system is

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For the purpose of deducing the arithmetical value of g, we have

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1 = .0042.

This, for reasons such as are assigned in Chap. XIII. must be an inexact value: but, it is not so enormously inexact, as the first resulting value (1−c) of the progression of the apogee: since, by a repeated process, g 1.0040105. The cause of the inexactness of the first resulting value, and the means of correcting it have been fully explained in the preceding pages: since what was there observed on the method of deducing the progression of the apogee is strictly applicable to the present case.

It is difficult to find any very simple mode of treating the Progression of the Apogee. Clairaut's method is as obvious as any other; and, as it has been observed, the preceding method in the text is analogous to it. But that, when the Regression of the Nodes is the object of investigation, is far from being the most simple method. Newton's is much more simple; and, which is a rare excellence, it at once shews the regression to be an obvious effect of the disturbing force, and affords the means of computing its quantity. This kind of excellence, however, depends in a great degree, on the nature of the subject of research, and consequently, in Physical Astronomy, is very limited. It cannot be expected to be found, (as the very terms, indeed,

signify) in abstruse subjects. But the method described in p. 250. although less simple than Newton's, has yet its peculiar advantages. g 1, which expresses the mean Regression, is equal to

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Now, as it has been before stated, the eccentricity (é) of the Solar Orbit is rendered variable by the action of the planets. It is subject to a Secular Equation; consequently the mean Regression of the Lunar Nodes is also subject to a Secular Equation. A similar inference was made in p. 181. from the value of 1c, relative to the Secular Equation of the Progression of the Lunar Apogee; and such inferences are more easily made from Clairaut's than from Newton's method.

Laplace has still a different method, but one resembling that which he uses for determining the Progression (pp. 212. 223. 2nde Partie, Liv. VII. Mec. Cel.): and it follows also immediately from this method that the Regression of the Nodes is subject to a Secular Equation.

The preceding pages relate principally to the periodical inequalities of the Moon: those inequalities which prevent her parallax, longitude, and latitude, being what they would be were the sole force acting on her the Earth's attraction. But the Progression of the Apogee, and the Regression of the Nodes, belong to a distinct class of inequalities, such as affect the very orbit itself, its dimensions and position in space. These inequalities are technically denominated the Variations of the Elements. One element is the position or longitude of the Apogee: another the longitude of the Node: the Variation of the former is the Progression of the Apogee, of the latter the Regression of the Node; and these have been treated of: at least, their mean quantities and the Secular Inequalities affecting them. A third element is the eccentricity of the Lunar Orbit. This, Newton, in the eleventh Section of his Principia, shewed to be subject to change from the Sun's disturbing force: and, on grounds and by considerations similar to those which we used in speaking of the cause of the Evection. But Newton gave no method of computing its quan

tity as he did in the case of the nodes. Indeed, the variation of the eccentricity is not, like the progression of the apogee, and the regression of the nodes, a distinct phenomenon: it is combined with, and influences, other inequalities. It could not, therefore, by the agreement of its computed and observed quantity, readily serve, like the two other variations, to confirm the Law of Gravitation.

A fourth element is the inclination of the plane of the Lunar Orbit. In his eleventh Section, Newton shewed the Variation of this element to be a necessary consequence of the Sun's disturbing force: and, in the third Book of the Principia he computes its law and quantity. A fifth element is the semi-axis of the Moon's Orbit, or her mean distance. If we look to phenomena that element has no variation. Newton, therefore, in the third Book of the Principia, could derive no confirmation of his principle and Law of Attraction from the agreement of its computed and observed quantity. The Moon's mean motion was found to be invariable, and therefore her mean distance would be so. Still it is remarkable that, the other elements varying, this should remain constant: that it should be so, both when the disturbing force acted, and when it did not. This is, in itself, a kind of phenomenon which requires an explanation. It is necessary to shew, at least for the purposes of curious inquiry, that a disturbing force can make no alteration in the mean distance. There cannot be a less self-evident proof of the Sun's attraction than the invariability of that element. It affords on first views, if any thing, a presumption against the principle of universal attraction. The invariability therefore, of the mean distance is a thing to be established on Newton's principles: and being established, is, at least, equally a proof of their truth as the Variability of the Apogee.

Newton has given nothing on this subject in his Principia: it was not to be expected that the founder of a great system should have had leisure to attend to all its details. Investigations of a nature so abstruse as those that have been just described, would, during the establishing of a new system, be postponed, and made to give place to others more obvious and important. The mathematicians, however, who succeeded Newton had leisure to attend to this subject.

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