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CHAP. XII.

Principle of the Method of correcting the Value of the Radius Vector, obtained by an Approximate Integration of the Differential Equa

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where (see pp. 98, &c.) represents the disturbing force, cannot

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generally be solved. assume that value of = 0; which value,

In order to approximate to its solution we

which is the integral of the equation when in other words, is the elliptical value of u, and the true value when no disturbing force acts. This value

Π

μ

is substituted in § (II = − − ), the equation integrated,

h2

and a new value of u obtained; which, since the conditions of the problem are rightly involved in the general differential equation, must be more nearly the true value than the one assumed.

Still it is not the true value : in order more nearly to approach to it, we may substitute the last obtained value in II, and again integrate the resulting equation. Now if we attend to the process of pp. 169, &c. we shall find that its effect is to add several small terms to the first assumed value of u. Suppose, (for the sake of stating the case in the most simple manner), that the first integration adds one small term to the value of u: then, if the process be repeated with this augmented value of u, II (see pp. 170, &c.) will contain more terms than it did before; which additional terms are entirely due to the augmentation of u: they may be viewed, therefore, as so many corrections to its value: and, accordingly, we need only compute the corrections to the value of II. Now this we can do by

the Table already formed (see p. 175.), for

a

cos. qv may re

present any term: either a deficient one in the elliptical value of u, or an additional one acquired by integration.

We have supposed the value of u to contain, after integration, one additional term: the fact is, it will contain several. The additional terms then in II will be corrections due to the additional terms of u: but, since these latter are, in the cases treated of, very small, we may deduce the corrections separately, one by one;

and

Q

--

a

cos. qv which may represent any term, will thus serve, by

repetition of process, to represent all.

This is a brief description of the principle of the method, which we will now exemplify.

The first principal additional term in the differential equation is (see p. 180.)

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If (see p. 100.) we divide this term by (2

2 m)2 - 1, then the result is an additional term in the value of u, or is a correction to its first assumed and elliptical value; and, if we equate

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In order therefore to find what new terms will be added to ПI, or what corrections to existing terms, we must, in the Table of p. 175. substitute for Q its preceding value, and 2 - 2 m for q.

The results then will be (see pp. 175, 176.), respectively,

A A

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Hence, there will be only one new term added to the value of II, the argument of which will be 4 v4 mv, and the term itself will be

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the corresponding additional term in the value of u, after integration, will be the preceding term divided by 16. (1—m)2 — 1.

The other parts are corrections of terms obtained by the first approximation and integration: first, since cos. O v = 1, the correction of the constant part of II will be (see lines 1, 3, 4, of this page),

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and the corresponding correction that would be given to u, after integration, is (see p. 100.) the preceding term with the sign changed.

Secondly, the term in the second line of this page, namely,

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is the correction of a term with the same argument already existing in II and the corresponding correction to the value of “,

resulting from integration, is the preceding correction divided by

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in the differential equation of p. 180: so that, if we were now to apply the correction, the coefficient of cos. cv would become

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and thus we may perceive that the correction due even to one additional term, has made a considerable alteration in the coefficient of that important term on which the progression of the apogee depends. But the coefficient will receive corrections from other additional terms.

For, as (see pp. 170, 176, &c.) L

a

cos.qu may represent any term

in the value of u, let us suppose the differential equation of p. 180. to be integrated, and that cos. qv represents the term whose

a

argument is the same as that of the second principal additional term of the equation: then, nearly,

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If this value of q be substituted in the Table p. 175. the arguments of the resulting corrections will be

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and of these corrections, there is only one that of which the argument is 4v-4mv-cv), which after a second integration, will produce a new term; the others are corrections of terms already obtained by the first integration. Now of these latter (and this is a point on which the determination of the progression of the Lunar Apogee depends) three have the argument c ", or serve to correct the coefficient of cos. cv: and their sum is

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cos. qv to represent the term which would be

added to the value of u, after the integration of the differential equation, and in consequence of that term therein contained, which has for its argument 2 v 2mv+co, then, as in the former case, there will result three corrections to the coefficient of cos. cv, and their sum will be

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in which expression, Q must equal the coefficient of

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(in the differential equation of p. 180), divided by

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We may conclude then from what has preceded, that the

terms

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