(1-a2)9

1

4x)4, (a 3a3) 40}

{A2 {2 + 3a2)2 + 5a2o(1 + ao) — 2(1 − a2)2}

5a(5 + 9a2)A ̧ + 5.7a2A1 }

3

{{(3+ 4a2)2 + 7a2(1+a2)−3(1 − a)2 } 43

- 7α (7+11a) A, +7.9.a As}

1

{{(4 + 5a2)2+ 9a2(1 + a2) − 4(1 − ∞2)2 } A ̧ (1 - a2)2

-9a (9+13a) As + 9.11.4},

&c.

5

&c.

456. By the aid of equation (120), it is easy to see that

457. The coefficient A, and its differences have a very simple form, when expressed in functions of B, for equations (121) and (126) give

A。 = (a22 + a2) B。

2aa'B1

0

- (a2+ a2) B2

3

2aa'B (a2+ a2)B,

5

&c.

&c.

458. The differences of A, and B, with regard to d are obtained from their differences with regard to a, for A, being a homogeneous function of a and a' of the dimension

1,

dA.

da'

&c.

da

&c.

Likewise B, being a homogeneous function of the dimension

dB.

da'

da

3Bi.

459. By means of these, all the differences of A, B, with regard to a', may be eliminated from the series R, so that the coefficients of article 449 become

a2

(i+1)

N2 = }{(2i+2) (2i+1) 4,41) — 2a (da) - a' (d)}

N1 = faa' Σ B(-1)

Naa' (B(i−1) + B(i+1)),

&c.

&c.

When i = 1, N1 = jaa' B. — § 2, and 3.7

a

a

a's

must be added to N.

12

460. The series represented by S and S' which are the bases of the computation, are numbers given by observation: for if the mean distance of the earth from the sun be assumed as the unit, the mean distances of the other planets determined by observation, may be expressed in functions of that unit, so that a = the ratio of the

a'

a

mean distance of m to that of m' is a given number, and as the functions are symmetrical with regard to a and a', the denominator of a may always be so chosen as to make a less than unity, therefore if eleven or twelve of the first terms be taken and the rest omitted, the values of S and S' will be sufficiently exact; or, if their sum be found, considering them as geometrical series whose ratio is 1 a2, the values of S and S' will be exact to the sixth decimal, which is sufficient for all the planets and satellites. Thus A, B, their differences, and consequently the coefficients Mo, M1, No, &c. of the series R are known numbers depending on the mean distances of the planets from the sun.

461. All the preceding quantities will answer for the perturbations of m' when troubled by m, with the exception of A,, which becomes

Jupiter's satellites, the equatorial diameter of Jupiter, viewed at his mean distance from the sun, is assumed as the unit of distance, in functions of which the mean distances of the four satellites from the centre of Jupiter are expressed.

CHAPTER VI.

SECULAR INEQUALITIES IN THE ELEMENTS OF THE ORBITS.

Stability of the Solar System, with regard to the Mean Motions of the Planets and the greater axes of their Orbits.

462. WHEN the squares of the disturbing masses are omitted, however far the approximation may be carried with regard to the eccentricities and inclinations, the general form of the series represented by R, in article 449, is

m'k. cos {i'n't- int + c} = R,

k and c are quantities consisting entirely of the elements of the orbits, k being a function of the mean distances, eccentricities, and inclinations, and c a function of the longitudes of the epochs of the perihelia and nodes. The differential of this expression, with regard to nt the mean motion of m, is

The expression dR always relates to the mean motion of m alone; when substituted in

It is evident that if the greater axes of the orbits of the planets be subject to secular inequalities, this value of a must contain terms independent of the sines and cosines of the angular distances of the bodies from each other. But a must be periodic unless i'n' in = 0; that is, unless the mean motions of the bodies m and m' be commensurable. Now the mean motions of no two bodies in the solar system are exactly commensurable, therefore