tan 0 = N sin (gt+6) + N, sin (g,t + 6,) + &c. or subtracting gt + from 0, tan(0-gt-6)= N1sin{(g,-g)t+6,−6}+N2sin{(5,−5)t+6,−6}+&c. If the sum of the coefficients N + N1 + N2 + &c. of the cosines in the denominator taken positively be less than N, tan (0 - gt - 6) never can be infinite; hence the angle gt 6 will oscillate 90°, so that gt +6 is the true motion of the m, and g = 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 r fig. 86. B 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 preceding 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 ySn gt+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" = gåt + 6, the longitude of the node n"; and so on for each disturbing body, the last circle will be the orbit of m. 2 507. Applying the same construction to h and 7 (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. 50s. 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 any diameter Da, and take the arc ac' = gt+6; on C' as a centre with radius equal to N,, de scribe 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 Ng, describe a circle, and having drawn C"a" parallel to Ca, take the a" C" get + 62, and so on. Let av Civ be the arc in the last circle, then if Cib be perpendicular to Ca produced, it is evident that Civbp, Cb = 9, and if CC be joined, tan =√ p2+q2, tan 0 = being the angle CivCb. 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. 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 period, and m'N' the orbit of a planet m', whose position with regard to EN is to be determined. By article 444, EA = q sin (n't + e') — p cos (n't + e') is the latitude of m above AN; 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 If EN be assumed to be the fixed plane at a given epoch, then p = 0, q = 0, but neither dp nor dq are zero; hence do (dp' dp). sin e' + (dq' dq) cos 0', dq) sin o' tan p', and substituting the values in article 498 in place of the differentials dp, dq, &c. there will result — {(1.2) - (0.2)} tan ø′′ sin (0′ — 0'') + {(1.3) — (0.3)} 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)2 + (q'−q)2 be two orbits supposed to be very small. be assumed as the fixed plane and the mutual inclination of the If the orbit of m at the epoch q=0, =0, 7=', p=0, 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 441, is 3m' + 2(a12-a2)2 {(a'2 + a2)S' + aa'S}e' cos (w' - w). If these be put in the value of de, rejecting the powers of e above the second, and if to abridge C= m'. na2. (2aS+3a'S') 3m' .na . { (a2 — 5a'2)aa'S+(a‘+6a2a'2 — 5a")S'}, 3m'.na'a' (2a'S-aS') h'e' sin w l'e' cos '; = C + C1 (h2 + l2) + C2 (hh' + ll') + C2 {(p' −p)2 + (q′ — q)2 — h12 — 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 y (p' − p)2 + (q'-q)= M2, a constant quantity. Again by article 483, 12 - g)t + 6, - 6} h2 + 12 = No + N,2 + 2NN, cos {(g, h'2 + 1⁄2 = N12 + N/2 + 2N'N' cos {(g, g) t + 6, −6} hh'+ll' = NN'+N,N,'+(NN',+N'N,) cos {g, -8)t + 6;−6}. |