with any sensible inequalities depending on the squares and product of the masses; for if the variation of be taken, considering all the elements as variable, the substitution of their periodic variations will make the whole vanish in consequence of the relations between the partial differences. 586. The longitude of the epoch is not affected by any variations of this order that are sensible in the planets, but they are of much importance in the theories of the moon and Jupiter's satellites. 587. The variations in the elements depending on the squares of the disturbing forces, are insensible in the theories of all the planets, except those of Jupiter and Saturn; they are only perceptible in the motions of these two planets, on account of the nearly commensurable ratio in their mean motions introducing the minute divisor 5n2n; therefore, if (dē), (dā), (dõ,) (dπ), (dÃ), (dē), be the secular variations in the elements depending on the second powers of the disturbing forces, and computed for the epoch from the equations in articles 580, and the two following, the equations (130) become, with regard to Jupiter and Saturn only, Whence the elements of the orbits of these two planets may be de termined with great accuracy for 1000 or 1200 years before and after the time assumed as the epoch. Periodic Perturbations in Jupiter's Longitude depending on the Squares of the disturbing Forces. 588. Where e2 is omitted, equation (97) becomes The eccentricity and longitude of the perihelion, when corrected for their periodic inequalities (175), and (185) (176) and (186), become, e + de + de and +, + Swa and the longitude of the epoch when corrected by its periodic variation, is e de,; by the substitution of these v becomes dv = (2e + 28e, + 2de,) sin {nt + e − w + de, — da, – dw2} : when the quantities that do not contain the squares of the disturbing forces are rejected, the developement of this expression is dv = {28e, + 2edw1.de {2ed,+ 2ede,.da, 2de.de} cos (nt + e − w); when the values of the periodic variations are substituted, the result 589. The radii vectores and true longitudes of m and m' in their elliptical orbits have been represented by r, r', v, v', are the periodic perturbations of these quantities, these two co-ordinates of m and m' in their troubled orbits, are r + dr, r' + dr', v + dv, v' + dv'. When these quantities are substituted in R becomes a function of the squares and products of the masses, it consequently produces terms of that order in the mean motion =-3ff.andt. dR having the factor (5n'-2n); they therefore form a part of the great inequalities in the mean motions of Jupiter and Saturn. A mistake has been observed in La Place's determination of these inequalities, which has been, and still is, a subject of controversy between three of the greatest mathematicians of the present age, MM. Plana, Poisson, and Pontécoulant, to whose very learned papers the reader is referred for a full investigation of this difficult subject. 590. The numerical values of the perturbations of Jupiter in longitude are computed from equations (159), (164), (172), (182), and (199), together with some terms depending on the fifth powers of the eccentricities and inclinations which may be determined by the same process as in the other approximations; his perturbations in latitude are computed from equations (160) and (177), and those in his radius vector from (158) and (163). 591. Hitherto the mass of the planet has been omitted when compared with that of the sun taken as the unit; so that half the greater axes has been determined by the equation a3 = its real value is found from 1+m = n2, or a = no (1 + fm) ; 1 whereas the semigreater axes of the orbits of Jupiter and Saturn ought therefore to be augmented by ma, m'a', quantities that are only sensible in these two planets. CHAPTER XI. INEQUALITIES OCCASIONED BY THE ELLIPTICITY OF 592. As the sun has hitherto been considered a sphere, his action was assumed to be the same as if his mass were united in his centre of gravity; but from his rotatory motion, his form must be spheroidal on account of his centrifugal force, therefore the excess of matter at his equator may have an influence on the motions of the planets. In the theory of spheroids it is found that the attraction of the redundant matter at the equator is expressed by Where is the ellipticity of the sun, the ratio of the centrifugal force to gravity at the solar equator, 7 the declination of a planet m relative to this equator, R' the semidiameter of the sun, his mass being unity. Therefore, the attraction of the elliptical part of the sun's mass adds the term to the disturbing action expressed by the series R in article 449. If this disturbing action of the sun's spheroidal form be alone considered, omitting y3, and substituting it gives, with regard to secular quantities alone, Thus the action of the excess of matter at the sun's equator produces a direct motion in the perihelia of the planetary orbits. 593. The effect of the sun's ellipticity on the position of the orbits may be ascertained from the last of equations (115), Since is the declination of the planet m on the plane of the sun's equator, if the equator be taken as the fixed plane, then will And if the eccentricity be omitted, F = (p — 34). 112 (22 — a'), a3 or substituting a. tan p. sin (nt + e − e) for z, |
