Motion of the Orbits of two Planets.

511. Imagine two planets m and m' revolving round the sun so remotely from the rest of the system, that they are not sensibly disturbed by the other bodies.

Let y = √(p'-p) + (q'-q)' be the mutual inclination of the two orbits supposed to be very small. If the orbit of m at the epoch be assumed as the fixed plane

and

=0, 7=', p=0, q=0,
tan' tan' y = p'2 + q's.

In this case, equations (140) and (128) become

Since the greater axes of the orbits are constant, the first shows that the inclination is constant, and the second proves the motion of the node of the orbit of m' on that of m to be uniform and retrograde, and the motion of the intersection of the two orbits on the orbit of m, in consequence of their mutual attraction, will be (1.0)t.

Secular Variations in the Longitude of the Epoch.

512. The mean place of a planet in its orbit at a given instant, assumed to be the origin of the time, is the longitude of the epoch. It is one of the most important elements of the planetary orbits, being the origin whence the antecedent and subsequent longitudes are estimated. If the mean place of the planet at the origin of the time should vary from the action of the disturbing forces, the longitudes estimated from that point would be affected by it; to ascertain the secular inequalities of that element is therefore of the greatest consequence.

The differential equation of the longitude of the epoch in article

3m'

+

4(a2-a2)2

2(a12-a2)9

.e

{(a'2 + a2)S' + aa'S}e' cos (' - @).

If these be put in the value of de, rejecting the powers of e above the second, and if to abridge

3m'. na3a' (4aa'S-(3a2-a'3) S') ̧

2.4.(a-a2)3

3m' . na. { (a2 — 5a'2)aa'S+(a‘+6a2a'2 — 5a')S'},
4. (a2-a2)

3m' .na3a' (2a'S—aS'),

de becomes

de

dt

But

le cos @,

l'e' cos w';

=

= C + C1 (h2 + l2) + C2 (hh' + ll')

+ C2 {(p' −p)2 + (q′ − q)2 — h22 — 1/2}.

513. This equation only expresses the variation in the epoch of m when troubled by m'; but, in order to have the effect of the whole system in disturbing the epoch of m, a similar set of terms must be added for each of the planets; but if the two planets m and m' alone be considered, their mutual inclination will be constant by article 511, hence y2 = (p' − p)2 + (q'−q)2 = = M2, a constant quantity.

Again by article 483,

h2 + 12 = N2 + N + 2NN, cos {(g, — g)t + 6, — 6} h'2+ l'2 = N12 + N2 + 2N'N cos {(g, - g) t + 6, −6}

hh'+ll' = NN'+N,N'+(NN',+N'N,) cos {g, −g)t + 6,−6}.

Substituting these in de, and to abridge, making

An C+C, (N2 + N,') + C2(NN' + N'N)
+ C(M-N'-N"),

B2C,NN'- 2C,N'N¡ + C(NN; + N,N'),

it becomes

de A. ndt + B cos {(g, g) t + 6, 6) dt.

514. The term Ant only augments the mean primitive motion of the planet m in the ratio of 1 to 1+ A'. so that the mean motion which should result from observation would be (1+ A')nt, corresponding to the mean distance

a

(1 + A`)}}

Knowing this distance, which is given by a comparison of the periodic times, the primitive distance a may be determined; but as A is an infinitely small fraction of the order of the masses m and m', this correction in the mean distance is insensible. The term Ant may therefore be omitted, so that the secular variation in the epoch

de=
sin {(g, g) t+6' — 6. }
gi g

(142)

The variation in the epoch, like the other secular inequalities in article 480, may be expressed in series ascending according to the powers of the time; but as the term depending on its first power is insensible, it will have the form

de = Ht+ &c.

This inequality is insensible for the planets; its greatest effect is produced in the theory of Jupiter and Saturn: but even then it is only de 0.0000006501.t for Jupiter, and for Saturn de'+0".0000015114.t, t being any number of Julian years from 1750. This inequality is not the 60th part of a sexagesimal second in a century, a quantity altogether insensible. Like all other inequalities it is periodic; but its period, which depends on g,-g, is for Jupiter and Saturn no less than 70414 years. The variation de, though of the order of disturbing forces, may, in the course of many centuries, become sensible, on account of the small divisor g,—g introduced by integration; but although it is insensible with regard to the planets, it is of much importance in the theories of the Moon and of Jupiter's Satellites.

Stability of the System, whatever may be the powers of the
Disturbing Masses.

515. The stability of the system has been proved with regard to the greater axes of the orbits, even when the approximation extends to the squares of the disturbing forces, and to all powers of the eccentricities and inclinations. Its invariability with regard to the other elements has only been proved on the hypothesis of the orbits being nearly circular, and very little inclined to each other and to the plane of the ecliptic; but as the same results may be derived from the general equations of the motion of a system of bodies, they equally exist whatever the eccentricities and inclinations may be, and when the approximation includes the squares of the disturbing forces, and they remain the same whatever changes the secular inequalities may introduce in the lapse of ages.

516. If the equations of the motion of a system of bodies in article 346 be resumed, and the equations in x, x', &c., multiplied respectively by

Em.y
S+Σm

Σm.

; m'y' - m'.

Στ . 1 . &c.
y

S+ Σm

;

·m'x' + m'.

Σm. x
S+ Σm

; &c.

their sum will be

Ση .

(xd2y — yd2x)

dt2

as may be seen by trial. The integral of the preceding equation is

A similar equation may be found in x, z, and y, z; and when 1, it will be found that

S+m'

dt

C, C, C", being constant quantities. Now ydx

area described in the time dt by the projection of the radius vector of m on the plane xy. This area on the orbit is √(1−e2); and if be the inclination of the orbit on the plane ry, cos √a(1-c2) is its projection. In the same manner

is the area described by the projection of the radius vector of m' on the same plane, and so on. preceding equations becomes

In consequence of these the first of the

m √a (1— e2) cos

= mm'

+ m' Ja' (1 − e'1) cos p' + &c. ydx' x'dy + y'dx — xdy') + &c. + C.

dt

If the elliptical values of x, y, x', y', be substituted, the first term of the second member of this equation must always be periodic; for, in consequence of the observations in article 466, the arcs nt, n't, never destroy one another in the expressions ydx', x'dy, &c. Hence, if periodic quantities and those of the fourth order be neglected, the last number of the equation is constant. If the products ydx', x'dy, &c., contained constant terms, they would be of the first order with regard to the masses; and as they are functions of the elliptical elements, their variation is of the second order; consequently, the variation of the terms mm'. y'dr, &c., is of the fourth order. If the periodic part of the values of the elliptical elements be substituted in the first member of the preceding equation, any terms resulting from that substitution that are not periodic will be of the third order, and may be regarded as constant. The second member of the equation in question may therefore be esteemed constant. Hence,