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The longitudes are estimated from the mean equinox of spring.

620. The series represented by S and S' in article 453 form the basis of the whole computation, but twelve or fourteen of the first terms of each will be sufficiently correct for all the planets.

The numerical values of the coefficients, A., A., &c. Bo, B., &c., and their differences, for Jupiter and Saturn, are obtained from the formulæ in article 455, and those that follow. The mean distances

of these two planets are, according to La Place,
a = 5.20116636, a' 9.5378709,

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621. These are given by the numerical values of equations (198), which are computed from the formula

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as the numerical values of all the quantities in these expressions are given, it is easy to find by their substitution, that

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where the digits 0, 1, 2, 3, &c. refer to Mercury, Venus, the Earth, Mars, Jupiter, Saturn, and Uranus.

622. By the substitution of the preceding data, equations (128) and (141), give the following results, when multiplied by the radius

reduced to seconds, or, by 206264".8, where is the sidereal

dw
dt

motion of the perihelion of Jupiter in longitude at the epoch 1750,

during a period of 3651 days: 2

tion of the centre: is the annual variation of the orbit of Jupiter

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do
dt

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of the ascending node of the orbit of Jupiter on the fixed ecliptic of 1750; and is the same variation with regard to the true ecliptic.

do

dt

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if then, the quantities (202) relating to Jupiter, be multiplied by

m√

a, those corresponding to Saturn will be found, and the form'√ a

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and by the substitution of the numerical values of article 613 and 615, it will readily be found, that in 1750

7 = 1° 15' 30" П= 125° 44′' 34",

y being the mutual inclination of the orbits of Jupiter and Saturn, and II the longitude of the ascending node of the orbit of Saturn on that of Jupiter. If the differential of these equations be taken and the numerical values of

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623. The variations in the elements that depend on the squares of the disturbing forces must now be computed, and for that purpose the numerical values of P, P', and their differences, must be found

from equations (165) and (166).

The coefficients Qo, Q1, &c., are given by the expansion of R,

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