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B. cos. (2

2 m v − c v),

A. cos. (2 v - 2 m v), C.cos (2v2 m v + c v), resulting from the first integration of the differential equation, would, by the repetition of the processes of approximation and integration, chiefly serve to correct the coefficient of cos. c v. This has been established by the very result of the process of Correction; but it is easy to perceive from an inspection of the Table of p. 175, that it must happen.

This is one important fact: another curious fact, in the preceding system of Corrections, is to be noted in the Correction which each term confers on itself. Thus, the term being

cos. qv,

a

the second correction of the Table of p. 175. is

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but (see p, 185.) that is the correction of II, and consequently the corresponding correction in the value of u resulting from integration will be

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but this being viewed as a new term in the value of u, the correction of the term involving cos. q v in II will be

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and the correction of the like term in the value of u resulting from a repeated integration will be (see p. 185, &c.),

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introduced by the approximate integration of the differential equa

tion, the more correct value of that term will be

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but the series of terms within the brackets is a geometrical series,

and accordingly if we make a

=

3 Ka
2h2. (1-q°)

=

(nearly),

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This expression represents the term involving cos. qv together with the whole series of corrections derived from itself: but the term is affected with other, besides the latter, corrections, although less important ones. The series of corrections is, in fact, interminable: for, every new term is a source of corrections which may be viewed as terms, and which, in that character, will give rise to ulterior corrections.

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3 K

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cos. qu: hence,

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.P. cos. pv be a term in the differential equation,

cos. pv, (a = 2(1 - p)

-

is the corrected term: let

then P', P", P", &c. represent those parts of the coefficients of

cos. (2 v

2 m v), cos. (2 v − 2 m v − c v), cos. (2 v —

2 mv + cv),

that are within the brackets (see p. 180.) the differential equation, with its partially corrected coefficients, will be

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These expressions for the coefficients are very convenient in computation, and give, very nearly, their true values; but not exactly so, since they embrace only the self-derived corrections. Now a term must serve to correct, besides itself, other terms.

For instance, the term

with the term

Q'

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a

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e.cos. (2 v − 2 mvcv), produce a correction of

the term involving cos. cv, (as may be seen in pp: 185. 187.); but this happens only when great exactness is required; for, the coefficient of the correction must involve the product of two small quantities, Q, for instance, and è. In like manner, if from the corrections of the term

m'
2h u3

(see pp. 177. 178.) we do not

exclude the corrections that involve the products of small quantities, there will arise, besides those we have stated, other corrections to the coefficients of cos. (2 v

for, since (see this page),

2 m v = cv) ;

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is the coefficient of cos. (2 v−2 m v) in the equation which is the integral of the preceding differential equation (p. 180. 1. 3.), it will be what Q represents in p. 185. and, accordingly, the correction derived from it will be

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.e [cos. (2v-2 mv − c v)+cos. (2 v−2 m v + c v)],

the terms, therefore, in the third and fourth lines of the preceding equation, will become

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If we refer to the Table of p. 175. it will be seen that the terms involving é, &c. require corrections similar to the preceding. Thus,

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being two terms of the value of II, or, which is the same, two terms in the differential equation, their corrections derived from themselves will be (see Table, p. 175.)

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We have now, almost enough for exactness, and certainly with sufficient fullness for the elucidation of method, deduced the several terms, and their corrections, of the differential equation, from which, by a previously established process of integration, (see pp. 99, &c.), u may be deduced. It is chiefly in the Lunar Theory that great accuracy is required: not that the determination of the Moon s place differs essentially, or in the analytical mode of treating it, from the determination of Venus's place disturbed by the Earth's action: for, both cases equally belong to the Problem of the Three Bodies. But, the Moon's irregularities carefully observed during a long series of years, and, from the circumstance of her proximity to the Earth, noted with superior exactness, furnish a surer and more eminent test of the truth of Newton's System, than the irregularities of any other planet. The test consists in the comparison of the Moon's computed with her observed place; if the one be accurately noted, the other must be scrupulously computed. The computation, however, after all, must be one of approximation. Some quantities must be rejected, and since by the operation of that peculiar process which is used (see pp. 162, &c.) the values of quantities are continually changing, there can be no general rule, founded on their mere minuteness, for the rejection of some and the retention of others. We cannot be sure of being correct by any method that is much short of actual trial.

But, if we could get rid of this class of difficulties, we should still have to contend with another arising from the necessary complication of the conditions of the problem The disturbance of the Elliptical System is no other than that of all its laws; and consequently it is their analytical expression which is subject to change. In the value of II, for instance, a term occurs (see p. 169.) —3us3, u2.

2

- 3

and s2 was assumed equal sin. gv; an assumption, in principle, not compatible with the existence of a disturbing force. Instead of s2 we ought to have assumed (s + òs)2 in which ds should be supposed to represent a variation of s arising from the disturbing

force. And this assumption would have introduced

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into

h2

the value of II. But ds, representing the variation of s from its

BB

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