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e2) cos &+m' √a' (1 − e'2) cos p' + &c.=C. (144)
are the areas described by the radius vector of m in the time dt, projected on the co-ordinate planes x z, and y z. But it is easy to see by trigonometry that the cosines of the inclination of the orbit on these planes are sin cos 0, and sin ø sin 0;
Similar expressions exist for all the bodies; and as the same reasoning applies to the two last equations (143), as to the first, they give
√a(1-e) sin cos 0 + m' √a' (1-e) sin p' cos 0'+&c. = C', m √a(1— eo) sin ø sin 0+m' √a′(1 − e'2) sin p′ sin 0'+=&c.C". (145)
518. These relations exist whatever the eccentricities and inclinations may be, and whatever may be the changes that they undergo in the course of ages from their secular inequalities, the approximation extending to the third order inclusively, and even to the squares of the disturbing forces.
519. A variety of results may be derived from them. Because
m √ a (e2+tan2)+¿m' √ a' (e's + tan2 p') + &c. = 2m √a +2m'√a' + &c. +2C.
But the last member is altogether constant: hence
m √a (e2+ tan2 $)+m' √a' (e'2+ tano p') + &c. = constant. It was shown that when the squares and products of the eccentricities and inclinations are omitted, the variations in the eccentricities
are the same as if all the planets moved in one plane; and that the variations in the inclinations are the same as if the orbits were circular, as these quantities vary independently of one another, e, e', &c., and , ', &c., may be alternately zero in the last equation, consequently, m√ a. e2 + m' √ a' . e12 + m' √a", e112 + &c. constant ; m√ a tan2 + m' √a'tan2p' + m' a" tan p" constant ; results that are the same with equations (136) and (140).
If quantities of the order of the squares of the eccentricities and inclinations be omitted, the tangents of the very small quantities , ', may be taken in place of their sines, so that by the substitution of
ptan sin 0,
in equations (145) they become
+ m' √ a' . q' + m" √a" q" + &c. = constant,
Ja. p + m' √ a' . p' + m'' √a" p" + &c. = constant.
520. Since the eccentricities and inclinations of all the orbits in the solar system are very small, the constant quantities in all the preceding equations of condition must be very small, provided the radicals √a, √a, &c., have the same signs, that is, if the bodies all move in one direction, which is the case in nature; it may therefore be concluded that the elements vary within very narrow limits.
521. Let there be only two bodies m and m', the mutual inclination of their orbits being
cos y cos cos + sin sin 'cos (e' then if the squares of the equations (144) and (145) be added, the result will be
Neglecting quantities of the fourth order, and putting all the constant quantities in the second member, it becomes
The constant in the second part of this equation is equal to the
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' +
2mm' Jaa'. ydy
= 0, (147)
for a and a' are constant. 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
mae2 + m'a'e2 2mm' a2a'2nn' √1 — e2 √1 — e cos y = constant, 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'2 + 2mm' . a2a'2nn'.
e/2 COS Y
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 e, e', siny; 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'a2a'2nn' = 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 sino 7 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
Tan I sin
Tan I cos =
m √a(1-e2) sinsino+m' √a'(1-e2) sin 'sin '+&c. ma(1-e) cos + m' √a'(1-e) cos '+&c.
m Ja(1-e2) sin cos 0+m'√a' ( 1 − e22)sinp' cos0'+&c. m√a(1-e) cos + m'√a'(1-e) 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