whence dr=(da-ase cos (nt+e-w)-aedw sin (nt+e-w))(1+2ecos(nt+e-w)) 3eda cos (nt + e−w)+2aede+ae (§5+de) sin (nt+e−w). If the values of da, de, dw, dh, de, from article 529, be substituted in this expression, after the reduction of the products of the sines and cosines to the cosines of multiple arcs, and substitution for Mo, M, N, N, Ns, from article 459, it becomes бр m' = E. C. cos i (n't — nt + e' — €), a 2 +m'. e. E.D. cos {i (n't nt + e' e) + nt + +m'. e'.E.E. cos {i (n't nt + e'e) + nt + where 537. Having thus determined the perturbations in the radius vec is known; and if substitution be made for da + m'e. E. G. sin {i(n't — nt + e' —e) + nt + e—w} + m'e'.Σ.H;. sin { i(n't — nt + e'—e)+nt + e − 538. In these values of dr and du, i includes all whole numbers, either positive or negative, zero excepted: dr and do will now be determined in the latter case, which is very important, because it gives the part of the perturbations that is not periodic. 539. If i 0 in the series R in article 449, the only constant term introduced by this value into dr will be Again, in finding the integral da the arbitrary constant a, that ought to have been added, would produce a constant term in dr. In order to find it, let the origin of the time be at the instant of the conjunction of the two bodies m and m', whence cos 0 = 1, and the first term of da in article 529 becomes where extends to all positive values of i from i = 1 to i = ∞. 540. If these values of dr and da be put in equation (148), the And as by article 392 the elliptical parts of r and v that are not periodic, or that do not depend on sines and cosines, are r = a, and vnte: those parts of the radius vector and true longitude that are not periodic are expressed by Thus the perturbations in longitude seem to contain a term that increases indefinitely with the time; were that really the case, the stability of the solar system would soon be at an end. This term however is only introduced by integration, since the differential equations of the perturbations contain no such terms; it is therefore foreign to their nature, and may be made to vanish by a suitable determination of the arbitrary constant quantities. In fact the true longitude of a planet in its disturbed orbit consists of three parts,—of the mean motion, of the equation of the centre, and of the perturbations. The mean motion of the planet is the only quantity in the problem of three bodies that increases with the time: the equation of the centre is a periodic correction which is zero in the apsides and at its maximum in quadratures; and the perturbations being functions of the sines of the mean longitudes of the disturbed and disturbing bodies are consequently periodic, and are applied as corrections to the equation of the centre. All the coefficients of these quantities are functions of the elements of the orbits, which vary periodically but in immensely long periods. The arbitrary constant quantities introduced by integration, must therefore be determined so that the mean motion of the troubled planet may be entirely contained in that part of the longitude represented by v. 541. The values of a, n, e, e, and, in the preceding equations, are for the epoch t = 0, and would be the elliptical values of the elements of the orbit of m, if at that instant the disturbing forces were to cease. Let not be the mean motion of m given by observation, then the second of the equations under consideration gives and let a, be the mean distance corresponding to n, resulting from the equation, If in this last expression n+n, n, and a + a, a, be put for - n, and a,, and if (n, — n)3, (a, − a), which are very small be omitted, then and substituting for n, it becomes a- = 2m'a 3n dA。 da and as a may be put for a, in the terms multiplied by m', the equations (150) become r + dr = a, - ¿ m'a; (dA;) v + dv = n,t + €. da Thus du no longer contains a term proportional to the time, and the mean motion of the disturbed planet is altogether included in the part of the longitude expressed by v, in consequence of the introduction of the arbitrary constant quantities n, and a,, instead of n and a. The part of or depending on the first powers of the eccentricities may be found by making i= 0 in the values of da, de, &c., in article 529; after which their substitution in dr of article 536, will give 542. If the different parts of the value of dr and dv be added, The periodic inequalities in the radius vector and true longitude of m -m'.e.f. cos (nt + e - w) - m'e'f' cos (nt + e-w') +m'.e.E.D. cos{i (n't nt + e' − e) + nt + e − w} + m'.e'. Z. E. cos {i (n't—nt + e − e) + nt + e −☎'}, + 2m'.e.f. sin (nt + c − w) + 2m'. e'.f". sin (nt + e-w') +m'.e.Z. G. sin { i (n't - nt + e − e) + nt + e − w} +m'.e'.Σ. H. sin { i (n't — nt + e' − e) + nt + e - w'}. The action of each disturbing body will produce a similar effect on the radius vector and longitude of m, and the sum of all will be perturbations in these two co-ordinates arising from the disturbing action of the whole system on the planet m. 543. It has been already observed that each of the periodic variations da, de, &c., ought to contain an arbitrary constant quantity a,, e,,,, &c., introduced by their integrations, so that their true values a, + da; e, + de; w, + dw; &c. &c. are Now, if the values of dr, dv, are to express the effects of the disturbing forces on the radius vector and longitude during a given time, these constant quantities must be so determined, that when t = 0, they must give e, + de = 0, w, + da = 0, &c. &c., as was done with da. Substituting these values in place of de, dw, &c., in equation (149), the resulting values will complete those of dr and dv, which will no longer contain any arbitrary quantity, but will express the whole change in the longitude and distance arising from the action of the disturbing forces. Hence, if (r) (v) be the elliptical values of r and v, given in article 392, but corrected for the secular |
