These inequalities are very considerable in the motions of the satellites in longitude. The whole then depends on the resolution of the equations (266) and (267); these, however, are not complete, as several terms arise from the perturbations depending on the squares and products of the disturbing forces. Action of the Sun depending on the Eccentricities. 836. The part of R depending on the action of the sun in the elliptical hypothesis is H being the eccentricity of Jupiter's orbit, and II the longitude of the perihelion; hence M.a.h cos (nt - 2Mte 2E + gt + I); and therefore, equation (265) becomes derdr a3 -M. H. cos (Mt+E-II) - 9M2. h. cos (nt - 2Mte 2E+ gt + r). contains rdr - 3N2. 3M3.h. cos (nt - 2Mt + € → 2E+ gt+r), .h.cos (nt +egt + r) whence by the method of indeterminate coefficients, the integral is which is the effect of the sun's action on the radius vector; and if it be substituted in equation (259), the perturbations in longitude depending on the same cause will be 837. The first term of the second number of this expression corresponds to the evection in the lunar theory, and is only sensible in the motions of the third and fourth satellites; but it is not the only inequality of this kind, for each of the roots g1, 52, 53, furnishes another. The perturbations corresponding to these for the other satellites are found, by reciprocally changing the quantities relative to one into those relating to the others. Inequalities depending on the Eccentricities which become sensible in consequence of the Divisors they acquire by double integra tion. 838. It is found by observation, that the mean motion of the first satellite is nearly equal to twice that of the second; and that the mean motion of the second is nearly equal to twice that of the third; n = 2n1, n1 = 2n2. or In consequence of the squares of these nearly commensurable quantities becoming divisors to the inequalities by a double integration, they have a very sensible effect on the preceding equations in longitude. 839. The only part of equation (259) that has a double integral is 3affndt.dR; and as the divisors in question arise from the angles nt - 2n,t, n ̧t – 2n ̧t alone, it is easy to see that the part of R containing these angles is, hcos (nt+egt - г) instead of ; and as by articles 828 and 826 ror a2 observing that n = 2n, nearly, the result will be a • {Fh + Gh,} . cos (nt-2n,t +e-2e, + gt + г), a which substituted in зaffndt. dR, and integrated, gives for the Again, since n, 2n, nearly, the action of m, on m, produces in Sv, an inequality similar to the preceding, which is An inequality of the same kind, and from the same cause, is produced also in the equation of the centre of m, by the action of m, for with regard to the inequalities we are now considering, article 574 shows m √ a sv m, va that δυ, π whence the inequality produced by the action of m on m, is Lastly, the action of m, on m. produces an inequality in m, analogous to that produced by the action of m on m,, which is therefore {G'h2+ 2 F'h,} sin (nt-2n,t+e-2€,+gt+r). a We shall represent the preceding inequalities by dv = - Q sin (nt-2n,te - 2, + gt + г) (268) (269) dv1 = + Q, sin (nt − 2n,t + e − 2€, + gt + г) These inequalities are relative to the root g, but each of the roots g1 g, ga, give similar inequalities in the motions of the three first satellites. No such inequality exists in the motion of the fourth satellite, since its mean motion is not nearly commensurable with that of any of the others. Inequalities depending on the Square of the Disturbing Force. 840. On account of the nearly commensurable ratios in the mean motions of the three first satellites the preceding equations must be added as periodic variations to the mean motions, as in the case of Jupiter and Saturn, by means of them several terms are added to equations (266) and (267), which determine the secular variations in the eccentricities and longitudes of the apsides. For if the eccen tricities be omitted, and μ = 1, the equations df, df' in article 433 relative to the planets, become 'dR dv 1) + a sin The secular variations with regard to the first satellite will be found by substituting R = (p-1) 3rs in the first of the preceding equations, and putting nt + e + dv for v, and a + 2ror for r; whence Then only attending to the terms depending on nt-2n,t +- 2€, and do given by (260) be substituted; and as da rodr if the values of a2 dA 2n−2n,~N} .sin (nt − 2n,t + e −26,) (p-1) n. a2 Since the mean longitudes nt + e and n,t + €, are variable, these angles must be augmented by the values of dv, dv,, in equations (268) and (269), so that nt + e + Q sin (nt − 2n,t + e − 2€, + gt + г) n,te,+Q, sin (nt - 2n,t + e -26, + gt + r) must be substituted in the sine of the preceding equation, which becomes, in consequence, |