485. Now let those of equations (131) that give then, in consequence of equations (132), and the preceding relations, = g { N lm √ a + N'l'm'√a' +N"l'"'m" √a" + &c. } ; if the preceding values of h, h', h", &c., l, l', &c., be put in this, a comparison of the coefficients of like cosines gives 0 = NN1m √a +N'N'1m' √ ̃a'+N"N", m" √a" + &c. 0 = NN ̧m √a +N'N' ̧m' √ a'+N'N",m' √ a" + &c. Again, if the values of h, h', h", &c., in equations (133) be respectively multiplied by they give Nm √a, N'm' √a', &c. Nmh √a+N'm'h' √a' + N"m"h" Na+ &c. (134) { Nom √a + N12m' √ā + N'2m'' √a'' + &c.} sin (gt+6), in consequence of the preceding relations. By the same analysis the values of l, l', l", &c., give Nmla+N'm'l' √ã + N"m"l'' √a" + &c. = { Nom √ ̃a + N'2m' √a' +N''1⁄2m'' √a" + &c.} cos (gt + 6). The eccentricities of the orbits of the planets, and the longitudes of their perihelia, are known by observation at the epoch, and if these be represented by e, e', &c. . ', &c. by article 481, h = é sin, h'e' sin ', &c., l = è cos ☎, l' = e' cos ☎', &c. ; therefore h, h', &c., l, l', &c., are given at that period. And if it be taken as the origin of the time t = 0, and the preceding equations give Tan 6= N.e sin.m√a+N'. e' sino'.m' √a' + &c. But, for the root g, the equations (132) give N' CN, N" C'N, N'"C"N, &c., = C, C', C" being constant and given quantities; therefore tan 6 = é sin m√a +C. e' sin '.m' √a' + &c. è cosa.m√ a +C. e' cos ☎'.m' √ a' + &c.' If these values of N', N", &c., be eliminated from equation (134), it gives N= Thus tan ē sin ☎m √ a + Ce' sin a'm' √a' + &c. {m√ a + C3m' √ a' + C'2m" √a"+&c. } sin 6 and N are determined, and the remaining coefficients N', N", &c., may be computed from equations (132), for the root g. In this manner the indeterminate quantities belonging to the other roots g, 52, &c., may be found. Thus the equations (133) are completely determined, whence the eccentricities of the orbits and the longitudes of their perihelia may be found for any instant t, before or after the epoch. 486. The roots g, g1, ge, &c., express the mean secular motions of the perihelia, in the same manner that n represents the mean motion of a planet. For example, the periodic time of the earth is about 365 days; 360° 3651 hence n = which is the mean motion of the earth for a day, and nt is its mean motion for any time t. The perihelion of the terrestrial orbit moves through 360° in 113270 years nearly; hence, for the earth, g= 360° 113270 = 19' 4".7 in a century; and gt is the mean motion for any time t; so that nte being the mean longitude of a planet, gt + 6 is the mean longitude of its perihelion at any given time. 487. The equations (133), as well as observation, concur in proving that the perihelia have a motion in space, and that the eccentricities vary slowly. As, however, that variation might in process of time alter the nature of the orbits so much as to destroy the stability of the system, it is of the greatest importance to inquire whether these variations are unlimited, or if limited, what their extent is. Stability of the Solar System with regard to the Form of the Orbits. 488. Because he sin o, le cos w, e2 = h2 + 12 ; in equations (133), the and in consequence of the values of h and e2 = N2 + N‚2 + N2 + &c. + 2NN; cos {(g, g) t + 6, − 6} + 2NN, cos {(g, − g)t + 62 − 6} + &c. (135) When the roots g, g,, &c., are all real and unequal, the cosines in this expression will oscillate between fixed limits, and e2 will always be less than (N + N, + N2 + &c.)2 = N2 + N,2 + &c. + 2NN1 + 2NN, + &c. taken with the same sign, for it could only obtain that maximum if (g1g)t + 6,- 6 = 0, (g2 — g)t + 6, — 6 = 0, &c., which could never happen unless the time were to vanish; that is, unless g1g0, g2 g 0, &c.; - = thus, if g, 81, 82, &c., be real and unequal, the value of e2 will be limited. 489. If however any of these roots be imaginary or equal, they will introduce circular arcs or exponentials into the values of h, h', &c., l, l', &c.; and as these quantities would then increase indefinitely with the time, the eccentricities would no longer be confined to fixed limits, but would increase till the orbits of the planets, which are now nearly circular, become very eccentric. The stability of the system therefore depends on the nature of the roots g, g1, g, &c.: however it is easy to prove that they will all be real and unequal, if all the bodies m, m', m', &c., in the system revolve in the same direction. 490. For that purpose let the equations de dt = 0.1 e' sin (' — ∞) + 0.2 e" sin (" — w') + &c. be respectively multiplied by me Ja, m'e' √a, m'e" Ja", &c., and added; then in consequence of the relations in article 484, and because sin (') sin (ww) = sin (')=sin(" the sum will be 0 = ede.m√a+e'de.m' √a'+e''de".m' √a" + &c.; and as the greater axes of the orbits are constant, its integral is e2m √a+e2m' √ a' + e'2m'' √a"' + &c. = C. (136) 491. The radicals a, a, &c., must all have the same sign if the planets revolve in the same direction; since by Kepler's law they depend on the periodic times; and in analysis motions in one direction have a different sign from those in a contrary direction: but as all the planets and satellites revolve from west to east, the radicals, and consequently all the terms of the preceding equations must have positive signs; therefore each term is less than the constant quantity C. But observation shows that the orbits of the planets and satellites are nearly circular, hence each of the quantities e2m √ a, e'am' √ a', &c. is very small; and C being a very small constant quantity given by observation, the first number of equation (136) is very small. As C never could have changed since the system was constituted as it now is, so it never can change while the system remains the same; therefore equation (136) cannot contain any quantity that increases indefinitely with the time; so that none of the roots g, g1, §2, &c., are either equal or imaginary. 492. Since the greater axes and masses are invariable, and the eccentricities are perpetually changing, they have the singular property of compensating each other's variation, so that the sum of their squares, respectively multiplied by the coefficients m √a, m' √a', &c., remains constant and very small. 493. To remove all doubts on a point so important, suppose some of the roots, g, 51, 52, &c., to be imaginary, then some of the cosines or sines will be changed into exponentials; and, by article 215, the general value of h in (133) would contain the term Cc, c being the number whose hyperbolic logarithm is unity. If Det, C'cat, D'est, &c., be the corresponding terms introduced by these imaginary roots in h, h', l', &c., then e2 would contain a term (C2 + D2) cat, e'2 would contain (C"2 + D2) cat, and so on; hence the first number of equation (136) would contain Cat {m √ a (C2 + D2) + m' √a' (C12 + D'2) + &c.}, a quantity that increases indefinitely with the time. If C be the greatest exponential that h, l, h', l', &c., contain, Cat will be the greatest in the first member of equation (136); therefore the preceding term cannot be destroyed by any other term in that equation. In order, therefore, that its first member may be reduced to a constant quantity, the coefficient of Cat must itself be zero; hence m √ a (C2 + D2) + m2 √ a' (C12 + D'2) + &c. = 0. But if the radicals a, a, &c., have the same sign, that is, if all the bodies m, m', &c., move in the same direction, this coefficient can only be zero when each of the quantities C, D, C', D', &c., is zero separately; thus, h, l, h', l', &c., do not contain exponentials, and therefore the roots of g, g,, &c., are all real. If the roots g and g, be equal, then the preceding integral becomes h = (b + b') C“ = (b + b') (1 + at Cat + + &c. 2 1.2 Thus the general value of h will contain a finite number of terms of the form Ct, which increases indefinitely with the time; the same roots would introduce the terms Dľ, C'ť, D't', &c., in the general value of 1, h', l', &c.; therefore the first member of equation (136) would contain the term t2 {m √ a (C2 + D3) + m' √ a' (C'12 + D'2) + &c.} ; and if t be the highest power of t in h, l, h', l', &c., t will be the highest power of t in equation (136); consequently its first member' can only be constant when m√ a (C2 + Do) + m' √ a' (C'2 + D'2) + &c. = 0, |