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 m2, m, 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 μ= 1.0055974; 893. μ determines the compression of Jupiter's spheroid, for Pμ.0.0217794, whence 10.0219012. If t be the time of Jupiter's rotation, T the time of the sidereal revolution of the fourth satellite, then is the ratio of the centrifugal force to gravity at Jupiter's equator. But az=25.4359, T = 16.689019 days; and, according to the observations of Cassini t = 0.413889 of a day, hence p = 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 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 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, hå, 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, 05".7h+35". 53h, +863".74 h, + (g- 2650". 1) h. (314) 898. As the motion of the apsides of the orbits of the satellites is almost entirely owing to the compression of Jupiter, in the first ap proximation the coefficient of h, may be made zero in equation (311); whence hence, if 10000" be put for g in equations (311), (312), (314), they and, by the substitution of these in equation (311), a still more approximate value of g will be found. This process must be continued till two consecutive values of g are nearly the same. In this manner it may be found that h, may be regarded as the true eccentricity of the orbit of the third satellite, and h, h,, h, are those parts of the eccentricities of the other three orbits, arising from the action of Jupiter's equator on the apsides of the third, and depending on g, 9399". 17, their annual and sidereal motion. 899. Again, if h and h, be made zero in equations (311) and (312), and g omitted in the divisor, then will g= 35114".7, g1 = 59152".3, and by the same method it will be found that In these h and h, are the real eccentricities of the orbits of the first and second satellites, and the other values, h, h1, h2, hs, &c., arise from the action of the other satellites corresponding to the roots g and gr 900. With regard to the inclinations of the orbits and the longitudes of their nodes, it appears, from article 892, that the system of inclinations for the root p, is 7, is the real inclination of the orbit of the second satellite on its fixed plane, passing between the equator and orbit of Jupiter; and 1, l, ls, are those parts of the inclination of the other three orbits depending on the root p,, and arising principally from the action of Jupiter's equator; for the attraction of that protuberant matter, by changing the place of the nodes of the second satellite, alters the relative position of the orbits, which changes the mutual attraction of the bodies, and produces the variations in the inclinations expressed by l, la, la; and it is for this reason that these quantities depend on the annual and sidereal motion of the nodes of the second satellite. 901. A similar system depends on each root of p, that is, on the annual and sidereal motions of the nodes of the orbits of the other three satellites. These are obtained from equations (307), &c.; for when the values of the masses and of u are substituted, they become 0=(p-185091") + 2998′′. 231, +1492".5+106".03 0=1772".61+(p−43214′′)+5610".4 +249". 4 l 183".447+1166".34+(p-9227".2)+813".7 20.47+81".09 +1272".81+(p-2650') . (315) 902. The first approximate value of p is found by making the coefficient of zero in the first of equations (315); whence p=185091"; and if this value of p be put in the three last of these equations divided |