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the Moon revolving round the Earth, it is the progression of the Lunar Apogee; 1 √(1 m) expounding its quantity. −

In the preceding instance then, but in that alone, there is a perfect coincidence of the assumed and resulting values of u: there are three assumed arbitrary quantities, and three equations. for determining them. If the disturbing force did not vary as the inverse cube of the distance, but as u", then (see pp. 123, &c.) the general differential equation will not assume the form

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except the eccentricity of the orbit be very small; or, which amounts to the same thing, the value of u, such as (see pp. 111, &c.)

u = a. cos. Nu + L,

on

will be only an approximate value. Moreover the value of N, which the motion of the apsides depends, determined by the preceding method (pp. 109, &c.) will be only a near value: or (to make the phraseology approach to a similarity with that of Newton's) the body's place can be found, by the fiction of a moveable ellipse, only in orbits that are very nearly circular *.

But, as approximate solutions must be resorted to, when exact ones cannot be obtained, Clairaut supposed that he should obtain one of the former kind, when on the ground and principle of the exact solution (see pp. 102. 197. &c.) he compared the assumed value of u with its value resulting from the integration of the differ ential equation, in which, account had been made of both parts of the disturbing force; that is, of the tangential as well as of that which acts in the direction of the radius. The value resulting from integration was (see pp. 145. 156.) of this form †.

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* Circulis finitimis, Newton, Princ. Sect.

The following forms which may be easily made to coincide with Clairaut's are yet not exactly his. See Theorie de la Lune, pp. 23, &c.

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which, compared with the former, gives three equations, (see p. 197.)

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from which (as before), the three arbitrary assumed quantities, a, cand e may be determined. The second equation ought, if the method were a right one, to determine c, and thence the progression of the Apogee.

Now if

e2= .0030107,

and (as it has been already shewn in p. 147.)

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or the progression of the Apogee in a whole revolution, will equal

1° 29′ 55′′ about half its real quantity, that is, half the quantity determined by observation *.

This, however, it may be said, is only an approximate solution and necessarily incorrect; because, during the computation, several quantities dependent on the square and cube of the eccentricity, on the eccentricity of the Solar Orbit, on the inclination of its plane to that of the Moon's, &c. are neglected. But we have shewn in pp. 181, &c. that no retention of such quantities and account made of them, can ever correct the preceding error. The real correction consists in repeating the integration of the differential equation,

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after the approximate value of II has been formed, not by the assumed value of u, but by the value that results from the first integration (see pp. 185, &c.) t.

* Clairaut having computed, according to the preceding method, the value of c, drew this conclusion: 'Donc ou l'attraction Neutonienne ne donne point ce vrai mouvement ou la solution precedente n'est pas propre a la determiner.' Clairaut, Theorie de la Lune, ed. 2. p. 27. He had before said, in the Memoirs of the Academy, 'Apres avoir mis a ce calcul toute l'exactitude qu'il demandoit, j'ai eté bien etonné de trouver qu'il rendoit le mouvement de l'apogee au moins deux fois plus lent que celui qu'il a par les observations: c'est a dire que la periode de l'apogee qui suivroit de l'attraction reciproquement proportionelle aux quarrés des distances seroit d'environ 18 ans, au lieu d'un peu moins de 9 qu'elle est reellement', and 'Une resultat aussi contraire aux principes de M. Newton me porte d'abord a abandonner entierement l'attraction. Mem. Acad. 1745. pp. 336, 354.

+ We have been very anxious to explain particularly and distinctly in what the real correction consists; because it is frequently stated, (one author copying after another) that Clairaut committed his first error by neglecting to take account of certain terms, or by not pushing the approximation far enough: whereas, as it has been shewn, (pp. 181, &c.) it was not the neglecting of terms, but the non repetition of the process of approximation that was the cause of the error.

CC

The real mode, however, of correcting the erroneous computation of the progression was by no means obvious. One proof of this is, that it eluded, for a time, Clairaut*, Dalembert and Euler, men of great sagacity and mathematical skill. For, as the Moon's Orbit was, very nearly, elliptical, the assumed elliptical value of the radius could not differ considerably from the resulting value. It seemed probable then that the comparison of the coefficients of like terms, which, in a simple hypothetical case, gave exact results, would, in this, give results nearly exact.

But, as it has been observed before, mere probabilities in such cases either determine nothing, or are fallacious. What is true of other terms is not so of that term which involves cos.cv. The peculiarity of its formation subjects it to a class of corrections from which the former are exempt.

These corrections have been given in pp. 185, &c. and Clairaut, in extricating himself from those embarrassments into which his first error had thrown him, shewed that he could correct, almost, completely, that error by taking account of the correction which the term

L'e.cos. (2 v

a

would introduce.

--

2mo - co),

This undoubtedly is, when numerically expounded, the greatest correction which the coefficient of cos. cv receives. It is (see p. 188.)

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* Thomas Simpson, the ablest Analyst (if we regard the useful purposes of Analytical Science) that this Country can boast of, affirms in the Preface to his Tracts, that he himself, previously to any communication with M. Clairaut, found that the motion of the Apogee could be accounted for on the received Law of Gravitation.

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If we use, therefore, this correction, we shall have, instead of the equation of p. 157. the following

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The correction, therefore, arising from one additional term is a little more than equal the term to be corrected: and this un

*If c were really = 1, the progression of the Apogee would be nothing but we are compelled, as in like cases, for the sake of approximation, to assume it at first of this value: for, c the quantity sought is involved in the expression of its value: we assume it, therefore, in the latter, of some determinate value, in order to escape from a vicious circle. The Science of Calculation abounds with such instances. If instead of c = 1, we had assumed it = .99154801, which we know from other sources to be its value, then, instead of the coefficient being

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faulty assumption of c = 1, in the involved expression for its value, in

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+ Clairaut's correction (Theorie the term 2 e cos. (2v- 2 m v have deduced; not exactly, since in finding the variation of

de la Lune, ed. 2. pp. 27, &c.) for cv), is nearly the same as what we

m'

23

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2h2

cos. 2 w, he neglects to take account of the variation of cos. 2 w. And the authority of Clairaut has served to entail this error on some subsequent authors.

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