by l, values of will be found; and when these last quantities are put in the first of equations (315), a new and more correct value of p will be found: by repeating the process till two consecutive values of p nearly coincide, it will be found that is the inclination of the first satellite on its fixed plane, arising chiefly from the attraction of Jupiter's equator, and given by observation; and 1, la, l, are the parts of the inclination of the other three orbits depending on p, the annual and sidereal motion of the nodes of the first satellite. 903. The third and fourth roots of p will be obtained by making the coefficients of 4 and 4, respectively zero in the third and fourth of the preceding equations; and, by the same method of approxima tion, it will be found that where l, and I are the real inclinations of the third and fourth satellites on their fixed planes, given by observation. 904. It now remains to compute the quantities depending on the displacement of Jupiter's equator and orbit, namely, the four values of A, e''L+ bt, and y''pt at - The first are found by Τ the substitution of the numerical values of the masses and of p-, in equations (285). Whence As A, B, C, are the moments of inertia of Jupiter's spheroid, assumed to be elliptical, the theory of spheroids gives and by observation, it is known that Jupiter's rotation is performed in 0.41377 of a day; and that his sidereal revolution is 4332.6 then, by the substitution of the numerical values of the other tities, all of which are given, it will appear that 'P =3".2007. quan By observation, the inclination of Jupiter's equator on his orbit was, in 1750, L = 3°.09996, and as are given by the theory of Jupiter at that epoch, which is nearly the annual precession of Jupiter's equinoxes on his expresses the longitude of the descending node of II will be the longitude. orbit. Jupiter's equator on his orbit, 180° of his ascending node; consequently sin (v + y') = sin (v – II). By observation, it is known that, in the beginning of 1750, whence II 313°.7592; y' = 46°.241 -† 0′′.2676 . t; and, with the preceding value of e', it will be found that 905. It appears, from observation, that the two first satellites move in circular orbits, and that the first moves sensibly on its fixed plane, from the powerful attraction of Jupiter's equator; consequently h and h,, corresponding to the roots g and g,, are zero, as well as the inclination 1, depending on the root p. Hence the sys tems of quantities in articles 899 and 902 are zero; and as, by observation, the real equations of the centre of the third and fourth satellites are = = 2h, 245".14, 2h, 553".73, 2h, 3002".04; and the real inclinations of the second, third, and fourth on their fixed planes, are =-1669".31, 1=-739′′.98, 1-897".998. By the substitution of these quantities in the different systems, Ch2, Chз, &c. &c. it will be found that the equations in articles 835 and 878, 832°.6825. sin (v + 46°.241-49".St) 897.998. sin (v3 + P3t + ^3) 145".45. sin (v + pat + ^3) 3 906. The following data are requisite for the complete determination of the motions of the satellites, all of them being estimated from the vernal equinox of the earth; the epoch being the instant of midnight, December 31st, 1749, mean time at Paris. The longitudes of the epochs of the satellites, estimated from the vernal equinox, were Longitudes of the lower apsides of the third and fourth satellites. г2 = 309°.438603 The values of g2, 53, &c., p, p,, &c., are referred to the vernal equinox of Jupiter; but in order to refer them to the vernal equinox of the earth, the precession of the equinoxes, 50", must be added to the first and subtracted from the second; and as all the quantities in question have already been given, it will be found that the annual and sidereal motions of the apsides were Also the annual and sidereal motion of Jupiter's equinox, with regard to the vernal equinox of the earth, is 49".8. The longitude of Jupiter's vernal equinox at the epoch was 46°.25, consequently y' = 46°.25 + t. 49′′.8, and the eccentricity of Jupiter's orbit at the epoch was ē 19831".47. In order to abridge g2t+F2, gat + г3, pt + ▲, &c., will be represented by wa, W3, N, N1, N2, N3. Theory of the First Satellite. Longitude. 907. Since h and h, are zero, equations (302) give only the two following values of Q; If equation (296) and the first of equations (316) be added to this, observing that 2nt + 2e 2ngt 2€, 180° + 3nt + 3e 3n,t it will be found that the true longitude of the first satellite in its eclipses, is for in the eclipses of the satellites by Jupiter, or of Jupiter by the satellites, the longitudes of both bodies are the same; the Earth, Jupiter, and the satellites being then in the same straight line, consequently MtEnt + E, U = v, consequently the term depending on the argument 2 (nt-Mt+e-E) vanishes. Latitude. 908. By article 880 the action of the sun occasions the inequality 3M 8 Sn (L') sin (v2U-pt- A) |