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The formula for

1

which in pp. 295, &c.

(a2-2aa' cos. w +a2)}

was deduced for the conveniently determining of the differential

dA dB
da' da'

coefficients Indeed, as it is plain, it is requisite for expanding the fourth

&c. will now serve another purpose.

term.

By means of it, then, and of the expanded forms of Chap. XVII. we may express R in a series of cosines of the mean motions, &c. dR dR and thence, immediately, the values of

> de' da

&c.

But in this research of the value of R it is a very important point to determine whether it contains any constant quantities. If there should be such, involving either the inclination, or the nodes, or the perihelion, &c. then, (see p. 420.) some of the elements would necessarily have secular variations.

In order to determine these constant parts in the value of R we must extend its development beyond the forms of pp. 276, 277, 279, and include terms involving e2, e, eé: but, it is not proposed to include terms involving higher powers or products: for, it is a supposition, in this as indeed it has been in all preceding enquiries, that e2, é2, ee', sin?. ? are very minute quantities. We

will examine the terms of the expression of R in their order; and, it will be convenient to premise, that we shall be principally guided in this examination by looking after terms the factors of which contain the cosines of similar arcs: for, it is plain, (see Trig. form [7], p. 27.) that one of the terms of

cos. (m t + a) cos. (m t + b),

when expanded must be constant and equal to ¦ cos. (a−b).

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Make p1, and the sixth term of the value of cos. (v — v), (see p. 275.) is constant and equal

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a

but the first term of, when expanded, must be ; there is, therefore, on this account, a constant term introduced equal to

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é

1—e cos. (nt + e − π) + 2 é'. cos. (n't + e − x')

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- 2 e é'. cos. (nt + e − π) . cos. (n't + e − π')
+ &c.

Now the third and fifth terms of the value of cos. (v v) (see p. 275.), are

- e cos. (n' t + e − π),

- e'. cos. (nt + e −

π),

which combined, respectively, with the third and second terms of

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thirdly, one of the cosines, resulting from developing the fourth

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ee'. cos. (n't

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which combined with the first term of the value of cos. (vv), (see p. 275.) produces

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Now, (see ll. 4, 14, 20.) these four constant cosines destroy each other consequently, at least up to

éé, &c. there are no constant terms in

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we may, therefore, combine the value of cos. (v'

m'r 22

with the value of - (see p. 410.)

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v) (p. 275.)

and deduce some

terms involving the cosines of constant arcs (π/ T): the coefficients, however, of such terms will, at the least, involve ee. But quantities (see p. 409.) involving sin2. ed are not to be taken

account of.

2

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= (if we take account merely of the constant parts)

2 dД e'2 d2 A e2

dA

A+ a.

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+a.

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2

+ a2. da 2

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d a2 4

d2 A é'2 'da 4

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+ &c.

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the constant part of P' will combine with the constant part of cos. (vv), which, see p. 275, is the sixth term of its value, and equal eé. cos. ( - ), and form a constant quantity. #), The constant part of P' (see p. 276.) is similar to the above value of P (1. 13.) and equal,

d B e2
B+ a.
+ &c.

but, for reasons already stated (see p. 409. 1. 17.) we need only reserve the first term B, which multiplied into m'ee cos. (π' — π) produces

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In deducing the other constant quantities of P' cos. (v' — v) we must proceed on the principle laid down in p. 409. 1. 22, &c. Now the third term [-e. cos. (n' t +ε' — π)] of cos. (v

bined with the third term

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da

v) com

Δά of P' (see p. 276.) when in

such term ▲ a' is expressed by its first term, namely, -a' é cos. U,

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and e. cos. (nt+e), the fifth term in the value of cos. (— v) (see p. 275.) combined, similarly to the above combination, with Aa, produces

d B

da

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Lastly, cos. (n't—nt + e' €), the first term of cos. (v - v), combined with

d2 B da.da

P'), when, instead of A a, Aa', their first terms

Aa. A a' (the sixth term of the value of

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These three last terms (11. 12, 16, 24.) then being multiplied by-m': the sum of the constant parts of

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m' P' cos. (v'—v) is

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There are no constant terms, within the prescribed limits (see p. 409.) to be derived from P'cos. (2v′ −2v), P'. cos. (3v' — 3v), &c. cos. (2 v′ — 2 v) (see p. 275.) contains no constant quantity: and the first constant quantity produced on the principle of p. 409. 1. 22, &c. is by the combination of its first term, namely, d2 B cos. (2 n't-2n1+2-2 e) with .Aa. A a', when for Aa, da.da'

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:

substituted but then the coefficient of the resulting term would involve e2 e'2.

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+(B'+ &c.) cos. (n't nt + e − e) + &c.

+ &c.

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the only constant term then, of which it is necessary to take account, is

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have

If F, then, be used to designate the constant part of R, we

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