Periodic Inequalities in the Radius Vector, depending on the Third Powers and Products of the Eccentricities and Inclinations. 630. These are occasioned by Saturn, and are easily found from equation (168) to be - 0.0003042733. cos (5n't - 2nt + 56′ — 2€ 12°.14694) + 0.0001001860. cos (5n't-2nt +5e' -2e+45°.27972)* Periodic Inequalities in Latitude. 631. These are obtained from equations (160) and (177). $ = 1°.3172, is the inclination of Jupiter's orbit on the fixed ecliptic of 1750, is the same, with regard to the variable ecliptic; is the longitude of the ascending node of Jupiter's orbit on the fixed ecliptic; = 6".4571, is its secular variation with regard to that de dt 14".6626 is its secular variation with regard to dt (80) = do dt the variable ecliptic. Equations (197) give -0.0000726, and (80) 0.0008113, for the variations depending on the squares of the disturbing forces; hence =- 0".078283, = 6".457, with regard to the fixed ecliptic, and de which are the only sensible inequalities in the latitude of Jupiter. Chap. XIV.] PERTURBATIONS OF JUPITER. 632. The action of the earth occasions the inequalities δυ Ξ { 0".000086. sin 2 (n't — nt + e' — €) in the longitude of Jupiter, n' being the mean motion of the earth, and the action of Uranus is the cause of the following perturbations in the longitude of Jupiter, where n' is the mean motion of Uranus. nt + 2e' w') 2nt + 3ε' — € - w') These are all the inequalities that are sensible in the motions of Jupiter; those of Saturn may be computed in the same manner. On the Laws, Periods, and Limits of the Variations in the Orbits of Jupiter and Saturn. 633. When the values of p, p', q, q', are substituted in equations (137) they give Whence, 9 N. cos (gt + 6) + N,. cos 6, q'N'. cos (gt + 6) + N,. cos 6,. (205) p'—p = (N'— N) sin (gt + 6); q'—q = (N'— N) cos (gt + 6), and at the epoch when t = 0 or in consequence of Nma+N'm' a' = 0 are given, all the constant quantities g, g,, 6, 6,, N, N', and N,, are obtained from the preceding equations. The variations in the inclinations are at their maxima and minima when gt+6 6, is either zero or 180°; hence if €, be substituted for gt + 6, equations (205) give for the maxima of the inclinations; and when 6, + 180° is put for gt+6, they give for the minima, The maxima and minima of the longitude of the nodes are given by the equations de = 0, de' = 0, or d.tan 0 = 0, and therefore pp' + qq′ = p2 + q, and by the substitution of the quantities in equations (205), it becomes N+N,. cos (gt + 6 − 6,) = 0, - If N, be greater than N independently of the signs, the nodes will have a libratory motion; but if N, be less than N, they will circulate in one direction. Tan√NN corresponds to the preceding value of cos (gt+661) ; it gives the inclination corresponding to the stationary points of the node. whereas the maxima and minima of the inclinations happen when The stationary positions of the nodes therefore do not correspond either to the maxima or minima of the inclination, or to the semiintervals between them. = p' = 0.04078 q' 0.01573, with these values, Mr. Herschel found tan = 0.02980. √1−0.43290. cos{21° 37′and for Saturn, tan Also tx 25".5756} 0.03287. √1+0.82665. cos (21° 37'-tx 25".5756}. N, N' 0.04442 N N' 0.01368; so that the maxima and minima of the inclinations of Saturn's orbit are 2° 32′ 40′′ and 0° 47′, and its greatest deviation from its mean state does not exceed 52′ 50′′. In Jupiter's orbit, the maximum is 2° 2′ 30′′, and the minimum 1° 17′ 10′′, and the greatest deviation from a mean state is 0° 22' 40'. The longitude of the node has a maximum and minimum in both orbits, because N, > N'. The extent of its librations in Jupiter's orbit will be 13° 9′ 40′′, and in Saturn's 31° 56′ 20′′, on either side of its mean station on the plane of the ecliptic supposed immoveable. The period in which the inclinations vary from their greatest to their least values, and the nodes from their greatest to their least longitudes, is by article 486 = 360° 360° 25".5756 = 50673 Julian years. 634. The limits and periods of the variations in the eccentricities and longitudes of the perihelia are obtained by a similar process, from equations (133), and those in article 485. The quantities he sin, l = e cos ∞, h′ = e' sin a', are known at the epoch, and equations (132) give l' = e' cos ☎', g=21".9905, |