By the same process it may be found, that where the periodic terms which are quite insensible are omitted, the secular variation in the longitude of the perihelion of Jupiter's orbit, depending on the squares of the disturbing force, including the equation (188), is

dP a'n3.t a' ̧

(de) = 3m". a°n' t bm√ a+2m'√ d'. {P (I + pdf )}

m'√ a

de

P

dP
de

e(5n' - 2n)1

583. Secular variations, depending on the squares of the disturbing forces, arise from the same cause in the mutual inclination of the orbits, and in the longitude of the ascending node of the orbit of Saturn on that of Jupiter. These are obtained from equations (178), considering the elements to be variable; then the substitution of their periodic variations will give, in consequence of

(dy)

3m12. aon3 ̧ m√ a+m'√ a' 5m√ a+ 2m'√ a'

.t.

(n)=

.t.

m√a+m'√ a' 5m√ a+2m'√ a'

{P (1) + P(dp)}

m'√ a'

(196)

d2 P

+

m √a+m' √ a'

m' √ a'

P

dP

(d) · ( && ) + ( P ) · (dp)+(1) · (&F')}·

de'

584. These are the variations with regard to the plane of Jupiter's orbit at a given time, but the variations in the position of the orbits of Jupiter and Saturn with regard to the ecliptic may easily be found, for p, p', being the inclinations of the orbits of m and m' on the fixed ecliptic at the epoch, and 0, 0' the longitudes of the ascending nodes estimated on that plane, by article 444,

and on account of the action and reaction of Jupiter and Saturn,

And from these four equations, it will readily be found, that

Thus when dy and y II are computed, the variations in the inclinations and longitude of the nodes when referred to the fixed plane of the ecliptic may be found.

585. The periodic variations in the eccentricities, inclinations, longitudes of the perihelia, and nodes, do not affect the mean motion

with any sensible inequalities depending on the squares and product of the masses; for if the variation of

be taken, considering all the elements as variable, the substitution of their periodic variations will make the whole vanish in consequence of the relations between the partial differences.

586. The longitude of the epoch is not affected by any variations of this order that are sensible in the planets, but they are of much importance in the theories of the moon and Jupiter's satellites.

587. The variations in the elements depending on the squares of the disturbing forces, are insensible in the theories of all the planets, except those of Jupiter and Saturn; they are only perceptible in the motions of these two planets, on account of the nearly commensurable ratio in their mean motions introducing the minute divisor 5n' 2n; therefore, if

(dē), (dão), (d7,) (dπ), (d), (80),

be the secular variations in the elements depending on the second powers of the disturbing forces, and computed for the epoch from the equations in articles 580, and the two following, the equations (130) become, with regard to Jupiter and Saturn only,

Whence the elements of the orbits of these two planets may be de

termined with great accuracy for 1000 or 1200 years before and after the time assumed as the epoch.

Periodic Perturbations in Jupiter's Longitude depending on the Squares of the disturbing Forces.

588. Where e is omitted, equation (97) becomes v2e sin (nt + e − @).

The eccentricity and longitude of the perihelion, when corrected for their periodic inequalities (175), and (185) (176) and (186), become,

e+de+de and +, + Swą,

and the longitude of the epoch when corrected by its periodic variation, is e + de; by the substitution of these v becomes

δει dv (2e + 28e, + 2de) sin {nt + e - w + de, dw, dw,}: when the quantities that do not contain the squares of the disturbing forces are rejected, the developement of this expression is dv = {28e, + 2edw1.de - e da} sin (nt + e −

@)

{2eda,+2ede,.da, 28e.de} cos (nt + e − w); when the values of the periodic variations are substituted, the result will be the inequality

3m2. aan3 5m √ a+4m2 √ a' ‚{P (d') + P' (LP)} ›

m' √a'

(2)

de

(5n'-2n)3

de

(5n'-2n)3

m' √ a'

589. The radii vectores and true longitudes of m and m' in their elliptical orbits have been represented by r, r', v, v',

are the periodic perturbations of these quantities, these two co-ordinates of m and m' in their troubled orbits, are

r + dr, r' + dr', v + dv, v' + dv'.

When these quantities are substituted in

R becomes a function of the squares and products of the masses, consequently produces terms of that order in the mean motion

=-3f.andt. dR