first member at a given epoch, for at that epoch all the elements are supposed to be known by observation; it ought, therefore, to be independent of the variation of the elements e, e', and y: its variation will be m√a. ede + m' √ a' e'de' +

for a and a' are constant.

2mm' Jaa'. ydy

= 0,

(147)

m√ a + m' √ a

This relation must always exist among

the secular variations of the eccentricities of the two orbits and their mutual inclination.

If the constant part of equation (146) be included in the second member it becomes

m2ae2 + m'a'e' 2mm' a2a'2nn' √T

-

ecos y con

stant, by the substitution of an and an' for √a and √a';

If this value be put in the preceding equation, and all constant quantities included in the second member, it becomes

m2. ae2 + m22. a'e' + 2mm' . a2a'2nn'.

1

COS Y

1 +

e2

C, being an arbitrary constant quantity.

C, is a very small quantity with regard to the squares and products of m and m', since they are multiplied by e2, e2, siny2; and that the mutual inclination of the two planes and their eccentricities are supposed to be very small, as is really the case in nature. Each term of the first member of this equation will therefore remain very small with regard to the squares and products of m and m'; if all the terms have the same sign, each term will then be less than C,. But because all the

planets revolve in the same direction round the sun, nt, n't, will have the same sign. Hence all the terms in the first member will be positive as long as y is less than 90°. But if y = 90°, then sin y = 1; cos y = 0, which reduces the equation to

m2ae2 + m22a'e'2 + 2mm'a ann' = C,

and the last term is no longer very small with regard to mm', which is impossible, since C, is very small with regard to the product of m and m', and that the other terms of the first member are positive. Thus, because the angle y never can attain to 90°, it follows that y, the inclination, and the eccentricities e, e', of the two orbits, will always be small; for, as cos y never can become negative, every term in the first member of the equation under discussion will be positive, and will remain very small with regard to the squares and products of the masses m and m'. That is to say, the coefficients e2 e' sin2 y will always remain very small, because they are small at present.

522. This reasoning would be the same whatever might be the number of planets, since each of them would only add terms to the first member of the equation under consideration, similar to those that compose it.

523. Thus it may be concluded that the planetary system is stable with regard to the eccentricities, the inclinations, and greater axes of the orbits, however far the approximation may be carried with regard to the elements of the orbits, even including the second powers of the disturbing forces.

524. La Place and Poisson have proved the stability of the solar system when the approximation extends to the first and second powers of the disturbing force, on the hypothesis that all the planets revolve in nearly circular orbits, little inclined to each other; but in a very able paper read before the Royal Society on the 29th April, 1830, Mr. Lubbock has shown that these conditions are not necessary in a system subject to the law of gravitation. He has obtained expressions for the variations of the elliptical constants, which are rigorously true, whatever the power of the disturbing force may be, whence it appears, that, however far the approximation may be carried, the eccentricities, the major axes, and the inclinations of the orbits to a fixed plane, contain no term that varies with the time, and that their secular variations oscillate between fixed limits in very long periods.

The Invariable Plane.

525. It has been already mentioned that in the motion of a system of bodies there exists an invariable plane, which, always retaining a parallel position, is easily found, because the sum of the masses of the bodies of the system respectively multiplied by the projections of the areas described by their radii vectores in a given time, is a maximum on that plane, and the sum of the projections on any other planes at right angles to it is zero. It is principally in the solar system that this plane is of importance, on account of the proper motions of the stars, and of the plane of the ecliptic, which render it difficult to determine the celestial motions with precision, this difficulty indeed is already perceptible, and will increase when very accurate observations, separated by very long intervals of time, must be compared with each other.

If I be the inclination of the invariable plane on the fixed plane which contains the co-ordinates x and y, and if be the longitude of its ascending node, by article 166.

and substituting the values of C, C', C", given by equations (144) and

(145),

Tan Isin

Tan I cos =

m√a(1-e) sino sino+m' √a'(1-e) sin 'sin '+&c.

m √a(1-eo) cos $+ m' √a'(1−e'2) cos p' + &c.

m √a(1-e) sin cos 0+m'√ a' (1 − e'2)sino' cose'+&c. m√a(1-e) cos + m'√a'(1-e'2) cos p'+&c.

The second members of these two equations have been proved to be invariable, even in carrying the approximation to the squares and products of the masses, whatever changes the secular variations may induce in the course of ages; and, by what Mr. Lubbock has shown, they must be constant, whatever the power of the disturbing force may be hence it follows, that the invariable plane retains its position, notwithstanding the secular variations in the elliptical elements of the planetary system.

526. The determination of this plane requires a knowledge of the

U

masses of all the bodies in the system, and of the elements of their orbits. Approximate values of these are only known with regard to the planets, but of the masses of the comets we are in total ignorance; however, as the mutual gravitation of the planets is sufficient to represent all their inequalities, it shows that, hitherto at least, the action of the comets on the planetary system is insensible. Besides, the comet of 1770 approached so near to the earth that its periodic time was increased by 2.046 days; and if its mass had been equal to that of the earth, it would have increased the length of the sidereal year by nearly one hour fifty-six minutes, according to the computation of La Place; but he adds, that if an increase of only two seconds had taken place in the length of the year, it would have been detected by Delambre, when he computed his astronomical tables from the observations of Dr. Maskelyne; whence the mass of the comet must have been less than the part of the mass of the earth. The same comet passed through the Satellites of Jupiter in the years 1767 and 1779, without producing the smallest effect. Thus, though comets are greatly disturbed by the action of the planets, they do not appear to produce any sensible effects by their reaction.

527. If the position of the ecliptic in the beginning of 1750 be assumed as the fixed plane of the co-ordinates x and y, and if the line of the equinoxes be taken as the origin of the longitudes, it is found that at the epoch 1750 the longitude of the ascending node of the invariable plane was : = 102° 57' 30", and its inclination on the ecliptic I 1°35′ 31′′; and if the values of the elements for 1950 be substituted in the preceding formulæ, it will appear that in 1950 N = 102° 57′ 15′′ ; I = 1° 35′ 31′′;

which differ but little from the first.

528. The position of this plane is really approximate, since it has been determined in the hypothesis of the solar system being an assemblage of dense points mutually acting on one another, whereas the celestial bodies are neither homogeneous nor spherical; but as the quantities omitted have hitherto been insensible, the position of the plane as it is here given, will enable future astronomers to ascertain the real changes that may have taken place in the forms and positions of the planetary orbits.

CHAPTER VII.

PERIODIC VARIATIONS IN THE ELEMENTS OF THE
PLANETARY ORBITS.

Variations depending on the first Powers of the Eccentricities and Inclinations.

529. THE differential dR relates to the arc nt alone, consequently the differential equation da = 2a2. dR in article 439 becomes

da + m'a2. in . ZA, sin i (n't

nt + e − e)

+ m'a en (i-1). M, sin {i(n't-nt + 6' — e)

-

+nt

+ nt +e-w} + m'a2e'n (i− 1). M] sin {i(n't – nt + e' − e) + nt + e-w'}. The integral of this equation is the periodic variation in the mean distance, and if represented by da, then

- m'a'e (i-1)n M, cos {i(n't-nt+e'-e)+nt+e-w}

i(n'-n)+n

m'a'e' (i-1)n M, cos {i(n't—nt+e'—e) +nt+e—w'}.

i(n'-n)+n

In a similar manner it may be found that the periodic variation in

3fandt dR is,

From the other differential equations in article 439 it may also be found that the periodic variation in the eccentricity is

Se m'a

n

i(n'-n)+n

M, cos {i(n't-nt+e'—e)+nt+e—w}