when t0, there will be a sufficient number of equations to determine all the quantities N, N,, &c. and 6, 6,, &c. 500. Also the roots g, g,, &c. are real and unequal, for if the equatjons (137) be respectively multiplied by √a. p; m√a.q; m' √a'. p' ; m' √a'. q, &c. and added, the integral of their sum will be (p2 + q2) m √ a + (p'2 + q'2) m' √ a' + &c. = C in consequence of the relations (0.1)m √ a = (1.0)m' √a' (139) Whence we may be assured by the same reasoning employed with regard to the eccentricities, that this equation neither contains arcs of circles nor exponentials, when the bodies all revolve in the same direction, so that all the roots are real and unequal. √{N2 + N2 + &c. + 2NN, cos {(g, − g)t + 6, — 6}+2NN, The expression √p2+q g)t + 6, − 6} + &c.}. is less than N+ N1 + N, + &c., on account of these coefficients being multiplied by cosines which diminish their values. The maximum of tan would be N + N, + &c., which it never can attain, since the differences of the roots g g, ga g are never zero; and as the inclinations of the orbits of the planets on the plane of the ecliptic are very small, the coefficients N, N1, &c., which depend on the inclinations, are very small also, and will always remain so. And the inclinations of the orbits will oscillate between very narrow limits in periods depending on the roots 8, 51, &c. 502. The plane of the ecliptic in which the earth moves, changes its position in space from the action of the planets, each producing a retrograde motion in the intersection of the plane of the ecliptic, and that of its own orbit; whence it appears, that if EN be the orbit of N S fig. 85. the earth at a given epoch, AN' will be its position at a subsequent period, and so on. The secular inequality in the position of the terrestrial orbit changes the obliquity of the ecliptic; but as it is determined from equations (138) it oscillates between narrow limits, never exceeding 3°, therefore the equator never has A coincided, and never will coincide with the E ecliptic, supposing the system constituted as it is at present, so that there never was, and there never will be eternal spring. 503. Since p2 + qo = tan30, p2 + q′′ = tan2p', &c. equation (139) becomes m√ a tan2 + m' √ a' tan'' + &c. = C. (140) Whence it may be concluded that the sum of the masses of all the bodies in the system multiplied by the square roots of half the greater axes of their orbits, and by the squares of the tangents of their inclinations on a fixed plane, will always be the same. If this sum be very small at any one period, and if all the radicals have the same sign, that is, if all the bodies revolve in the same direction, it will always remain so; and as in nature, the inclinations of all the orbits on the plane of the ecliptic are very small, and the bodies revolve in the same direction, the variations of the inclinations compensate each other, so that this expression will remain for ever constant, and very small. 504. Other two integrals may be obtained from the equations (137). For if the first be multiplied by m√a, the third by ní ✔a', the fifth by m" Ja", &c. &c. their sum will be in consequence of the relations in article 484, the integral of which is m√ā. p + m'√a' . p' + m” √a” p′′ + &c. = constant. In a similar manner the differential equations in q, q', give m√ a.q + m'√ a q′ + m"√a" q' + &c. = constant. 505. With regard to the nodes tan 0 = P, and substituting for p and q, tan 0 = N sin (gt+6) + N, sin (g,t + 6,) + &c. or subtracting gt + from 0, tan(0-gt-6)= Nsin{(g,-g)t+6,−6}+N,sin{(52−g)t+6,−6}+&c. If the sum of the coefficients N+ N1+Ng+ &c. of the cosines in the denominator taken positively be less than N, tan (0 — gt — 6) never can be infinite; hence the angle between +90° and gt 6 will oscillate 90°, so that gt + 6 is the true motion of the nodes of the orbit of m, and g = 360° As in gene ral the periods of the motions of the nodes are great, the inequalities increase very slowly. From these equations it may be seen, that the motion of the nodes is indefinite and variable. The method of computing the constant quantities will be given in the theory of Jupiter, whence the laws, periods, and limits of the secular variations in the elements of his orbit, will be determined. 506. The equations which give p, q, p' q' may be expressed by a diagram. Let An be the orbit of the planet m at any assigned time, as the beginning of January, 1750, which is the epoch of many of the fig. 86. French tables. After a certain time, the action of the disturbing body m' alone on the planet m, changes the inclination of its orbit, D and brings it to the position Bn. But m" acting simultaneously with m' brings the orbit into the position Cn m" acting along with the pre B A ceding bodies changes it to Dn", and so on. It is evident that the last orbit will be that in which m moves. So the whole inclination of the orbit of m on the plane An, after a certain time, will be the sum of the finite and simultaneous changes. Hence if N be the inclination of the circle Bn on the fixed plane An, and ySngt + 6 the longitude of its ascending node; N' the inclination of the circle Cn' on Bn, and Sn'g't + 6' the longitude of the node n'; N" the inclination of the circle Dn" on Cn', and ySn" get +6, the longitude of the node n"; and so on for each disturbing body, the last circle will be the orbit of m. 507. Applying the same construction to h and I (133), it will be found that the tangent of the inclination of the last circle on the fixed plane is equal to the eccentricity of the orbit of m; and that the longitude of the intersection of this circle with the same plane is equal to that of the perihelion of the orbit of m. 508. The values of p and q in equations (138) may be determined by another construction; for let C, fig. 90, be the centre of a circle whose radius is N; draw a fig. 87. C any diameter Da, and take the arc aC' = gt + 6; on C' as a centre with radius equal to N,, describe a circle, and having drawn C'a' parallel to Ca, take a'C' = g,t + 6, on C" as a centre with radius equal to N2, describe a circle, and having drawn C"a" parallel to Ca, take the a" C" get + 6, and so on. Let av Civ be the arc in the last circle, then if Cb be perpendicular to Ca produced, it is evident that Civbp, Cb = q, and if CCiv be joined, 509. The equations which determine the secular variations in the inclinations and motions of the nodes being independent of the eccentricities, are the same as if the orbits of the planets were circular. Annual and Sidereal Variations in the Elements of the Orbits, with regard to the variable Plane of the Ecliptic. 510. Equations (128) give the annual variations in the inclinations and longitudes of the nodes with regard to a fixed plane, but astronomers refer the celestial motions to the moveable orbit of the earth whence observations are made; its motion occasioned by the action of the planets is indeed extremely minute, but it is important to know the secular variations in the position of the orbits with fig. 88. N m' regard to it. Suppose AN fig. 88, to be the plane of the ecliptic or orbit of the earth, EN the variable plane of the ecliptic in which the earth is moving at a subsequent A period, and m'N' the orbit of a planet m', whose position with regard to EN is to be determined. E EA q sin (n't + is the latitude of m above AN; By article 444, e') − p cos (n't + e') and the latitude of m' above AN' is Am' q'. sin (n't + e') - p' cos (n't + e'). = As the inclinations are supposed to be very small, the difference of these two, or m'A EA is very nearly equal to m'E the latitude of m' above the variable plane of the ecliptic EN. If be the inclination of m'N' the orbit of m' to EN the variable ecliptic, and the longitude of its ascending node, then will tan . sin p' — p; tan cos e q' Whence tan = √(p'—p)2 + (q'−q)o___ tan 0' = If EN be assumed to be the fixed plane at a given epoch, then = 0, but neither dp nor dq are zero; hence dp = (dp' — dp) . sin 0' + (dq' — dq) cos 0', and substituting the values in article 498 in place of the differentials dp, dq, &c. there will result (0.2)} tan p" sin (0′ — 0'') + {(1.3) — (0.3)} |
