m' √(x' — x)2 + (y' − y)2 + (z' — z)2 m' (xx'+yy'+zz') p3 There will arise terms in R, multiplied by m'm", which will be functions of nt, n't, n''t, when substitution is made of the elliptical values of the co-ordinates; and as the mean motions cannot destroy each other, these terms will only produce periodic terms in dR. Should there be any terms independent of the mean motion nt in the development of R, they will vanish by taking the differential dR. And as terms depending on nt alone will have the form m'm". dP, P being a function of the elliptical co-ordinates of m., there will arise in fd. R terms of the form m'm"fdP = m'm" . P, since dP is an exact differential. These terms will then be of the second order after integration, and such terms are omitted in the value of this function. The variation of the co-ordinates x, y, z, produced by the action of m" on m only introduce into the preceding part of R terms multiplied by m'm" and functions of the three angles nt, n't, n't; and as these three mean motions cannot destroy each other, there can only be periodic terms in dR. The terms depending on nt alone, only produce periodic terms of the order m'm" in dR. The same may be proved with regard to the part of R depending on the action of m" on m. 470. Hence whatever may be the number of disturbing bodies, when the approximation includes the squares and products of the masses, the variation of the elliptical elements of the disturbed and disturbing planets only produce periodic terms in dR. 471. Now the variation of =-3ffandt. dR is зanffdt. d. SR + 3aff (ndt. dR. SåR). It was proved in article 464 that d♪R = 0 in considering only secular quantities of the order of the squares of the masses. It is easy to see from the form of the series R that dRdR = 0 with regard to these quantities, consequently the variation of the mean motion of a planet cannot contain any secular inequality of the first or second order with regard to the disturbing forces that can become sensible in the course of ages, whatever the number of planets may be that trouble its motion. And as da = 2a dR becomes S da = {2a2fdR + 8a3f (dR ƒdR) }, by the substitution of (a + da2) for a2, da cannot contain a secular inequality if ♪ does not contain one. 472. It therefore follows, that when periodic inequalities are omitted as well as the quantities of the third order with regard to the disturbing forces, the mean motions of the planets, and the greater axes of their orbits, are invariable. The whole of this analysis is given in the Supplement to the third volume of the Mécanique Céleste; but that part relating to the second powers of the disturbing forces is due to M. Poisson. Differential Equations of the Secular Inequalities in the Eccentricities, Inclinations, Longitudes of the Perihelia and Nodes, which are the annual and sidereal variations of these four elements. 473. That part of the series R, in article 449, which is independent of periodic quantities, is found by making i = 0, for then and if the differences of A, A, with regard to a' be eliminated by their values in article 458, the series R will be reduced to + dF dp dF dq {e2 + e12 − ( p' − p)2 — (q' — q)2 } ee'. cos ( ☎) ; 3m' (aa'S+ (a2 + a')S'), ee'. sin (w' — ∞) 2 (a12- a2)2 3m'aa'S' 4(a12 — a2)2 e 3. m' (aa'S + (a2 + a13)S'), e' . cos (☎' — ☎) 474. When the squares of the eccentricities are omitted, the differential equations in article 441 become an dF e da dt an dF e de dq dF --an. dq dt dp dF If the differentials of F, according to the elements, be substituted 475. But tan = √ p2 + q2 and tan 0 = q P, and when the squares of the inclinations are omitted cos = 1, dp = dp sin 0 + dq cos 0; d0 = hence and substituting the preceding values of dp, dq, the variations in the inclinations and longitude of the node are, 476. The preceding quantities are the secular variations in the orbit of m when troubled by m' alone, but all the bodies in the system act simultaneously on the planet m, and whatever effect is produced in the elements of the orbit of m by the disturbing planet m', similar effects will be occasioned by the disturbing bodies m', m", &c. Hence, as the change produced by m' in the elements of the orbit of m are expressed by the second terms of the preceding equations, it is only necessary to add to them a similar quantity for each disturbing body, in order to have the whole action of the system on m. The expressions (0.1), [0.1 have been employed to represent the coefficients relative to the action of m' on m; for quantities relative to m which has no accent, are represented by 0; and those relating to m' which has one accent, by 1; following the same notation, the coefficients relative to the action of m" on m will be (0.2), [0.2 ; those relating to m'"' on m by (0.3), 0.3; and so on. Therefore the secular action of m" in disturbing the elements of the orbit of m will be 0.2 e" sin ("— w); (0.2) |0.2 = cos (∞" — ∞) - cos (0 - 0"). (0.2) tan sin (00"); (0.2) + (0.2) tan " 477. Therefore the differential equations of the secular inequalities of the elements of the orbit of m, when troubled by the simultaneous action of all the bodies in the system, are de dt = 0.1 e' sin (w) + 0.2 e'' sin (w" → ∞) аф dt de = (0.1) tan o'sin (0—0') + (0 . 2) tan p" sin (0-0")+&c. -= • {(0 . 1) + (0 . 2) + &c. } +(0 . 1) dt 478. All the quantities in these equations are determined by observation for a given epoch assumed as the origin of the time, and when integrated, or (which is the same thing) multiplied by t, they give the annual variation in the elements of the orbit of a planet, on account of the immense periods of the secular inequalities, which admit of one year being regarded as an infinitely short time in which the elements e,, &c., may be supposed to be constant. 479. It is evident that the secular variations in the elements of the orbits of m', m", m'", &c., will be obtained from the preceding equations, if every thing relating to m be changed into the corrcsponding quantities relative to m', and the contrary, and so for the other bodies. Thus the variation in the elements of m', m", &c., from the action of all the bodies in the system, will be &c. .e. sin (') + 2.1. e'. sin ('-w")+&c. do' = (1.0) + (1. 2) + &c. dt |