The numerical values of F, G; F', G', are determined from articles 825, 826, and 827, to be and with the same quantities the coefficients Q, Q1, Q2, of the tions in article 839 are found to be equa (302) Not only these quantities, but several data from observation are requisite for the determination of the unknown quantities from equations (271) and (290). 890. The eclipses of the third satellite show it to have two distinct equations of the centre; the one depending on the apsides of the fourth satellite is 2h, = 245".14. The other datum is the equation of the centre of the fourth satellite, which is, by observation, equal to 3002".04 2h,. Again, observation gives the annual and sidereal motion of the apsides of the fourth satellite equal to 2578".75, which, by article 831, is one of the four roots of g in equation (271), so that g, 2578".75. And, lastly, observation gives 43374" for the annual and sidereal motion of the nodes of the orbit of the second satellite on the fixed plane, which is one of the roots of p in equation (290), so that P, 43374". 891. If the values of m, and m, as well as all the quantities that precede, be substituted in equations (271) and (290), they become, when the first are divided by h,, and the last by s {80409 +179457 μ.+51581′′.5 m + 1686′′.44 m, + -26505".7 m2 + 45344".8 mmą 19393".4 m2 +544.86 m2 + 69".16 m3. 0={18305".3m+72999".2 m2 -63180".4 mm,} 2511.6-35317". μ-14128 m-13455". m.-584′′.554 m ̧ + 594".41 m". + 256′′.12 m2 · 677′′.04 mm. + 592′′.6 m1⁄2§. +{1352′′.6 − 1569′′. m+1342". m.) (306) m2} h h1 0-4831".9 m. 0=2998".23+(40342′′.3-179457′′. μ-1686′′.44m-248′′.57 m2) — 892. These are the particular values of equations (271) and (290) corresponding to the roots g, 2578".82, and p, 43374" alone. By the following method of approximation, nine of the unknown quantities are obtained from these eight equations, together with equation (300). The inclinations of the satellites are very small, and the two first move nearly in circular orbits, therefore the quantities are so minute that they may be made zero in the equations (303), (306), (307), in the first instance; and if m be eliminated by equation (300), these three equations will give approximate values of the masses m1⁄2, m1, and of μ, and then m will be obtained from equation (300). But, in order to have these four quantities more accurately, their approximate values must be substituted in equations (304), (305), (308), (309), and (310), whence approximate values of will be found. Again, if these approximate values of be substituted in equations (303), (306), and (307), and if m be eliminated by means of equation (300), new and more accurate values of the masses and of μ will be obtained. If with the last values of the masses and of the same process be repeated, the unknown quantities will be determined with still more precision. This process must be continued till two consecutive values of each unknown quantity are nearly the same. In this manner it is found that 1.0055974; m0.173281; m2=0.884972; } = 0.00206221h; h1 = 0.0173350. h.; h2 0.0816578 . hg; m,= 0.232355; 1 = 0.0207938 1,; l2 = 0.0342530 7,; 13 =- 0.000931164 1, 893. μ determines the compression of Jupiter's spheroid, for If t be the time of Jupiter's rotation, T the time of the sidereal revolution of the fourth satellite, then T = is the ratio of the centrifugal force to gravity at Jupiter's equator. But a = 25.4359, T = 16.689019 days; and, according to the observations of Cassini t = 0.413889 of a day, hence $ = 0.0987990, and p = 0.0713008. As the equatorial radius of Jupiter's spheroid has been taken for unity, half his polar axis will be 1 - p = 0.9286992. The ratio of the axis of the pole to that of his equator has often been measured the mean of these is 0.929, which differs but little from the preceding value; but on account of the great influence of the matter at Jupiter's equator on the motions of the nodes and apsides of the orbits of the satellites, this ratio is determined with more precision by observation of the eclipses than by direct measurement, however accurate. The agreement of theory with observation in the compression of Jupiter shows that his gravitation is composed of the gravitation of all his particles, since the variation in his attractive force, arising from his observed compression, exactly represents the motions of the nodes and apsides of his satellites. 894. If the preceding values of the masses of the satellites be divided by 10,000, the ratios of these bodies to that of Jupiter, taken as the unit, are 895. Assuming the values of the masses of the earth and Jupiter in article 606, the mass of the third satellite will be 0.027337 of that of the earth, taken as a unit. But it was shown that the mass of the moon is of that of the earth. Thus the mass of the third satellite is more than twice as great as that of the moon, to which the mass of the fourth is nearly equal. 896. In the system of quantities, S32578".82 h – 0.00206221 h = 6,3 ha h = 0.0173350 h = 6,3 h (3) ha=0.0816578 h = C ̧3) hs h, may be regarded as the true eccentricity of the orbit of the fourth satellite, arising from the elliptical form of the orbit, and given by observation. And the values of h, h1, h, are those parts of the eccentricities of the other three orbits, which arise from the indirect action of the matter at Jupiter's equator; for the attraction of that matter, by altering the position of the apsides of the fourth satellite, changes the relative position of the four orbits, and consequently alters the mutual attraction of the satellites, and is the cause of the changes in the form of the orbits expressed by the preceding values of h, h1, h.. This is the reason why these quantities depend on the annual and sidereal motion of the apsides of the fourth satellite. 897. A similar system exists for each root of g, arising from the same cause, and depending on the annual and sidereal motions of the apsides of the other three satellites. These are readily obtained from the general equations (271), which become, when the values of the masses and of the quantities in equations (301) are substituted, |
