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HISTOEY OE PHYSICAL ASTBONOMY.

the disturbing satellites; hut the results he obtained in this instancewere by no means so accurate as his estimation of the mass of the secondsatellite.

While Bailly was engaged in these researches, the Academy of Sci-ences offered their prize of 1766 for an investigation of the inequalitiesof the satellites. The successful competitor was Lagrange, who trans-mitted to the Academy a magnificent memoir on the subject. This illus-trious geometer grappled with the real difficulties of the problem, whichit must be acknowledged that Bailly left untouched. When the orbits ofthe satellites are supposed to be circular, and their planes to be all coin-cident, the effects of their mutual attraction may be computed in any in-stance by considering in succession the action of each body upon the dis-turbed satellite. For this purpose the method of approximation employedin the lunar theory is amply sufficient, and by means of it the inequalities,independent of the eccentricities and inclinations, may be easily calcu-lated. But when the actual conditions of the orbits are taken into ac-count, the mutual perturbations of the satellites become so entangledtogether as to render the ordinary method of integration totally inappli-cable, and it becomes absolutely necessary to devise some adequate meansof surmounting the difficulties of the problem. This important step wasaccomplished by Lagrange, who investigated the inequalities of eachsatellite by a method which embraced the simultaneous actions of thesun and the other three satellites, as well as the disturbance arising fromthe oblate figure of Jupiter . This comprehensive mode of treating the sub-ject entailed upon him an extent of analytical research which, to useDelambre’s words*, it is somewhat frightful to contemplate. Neglectingfirst the eccentricities and inclinations, he found for each of the threeinterior satellites an inequality in longitude of 437 days, correspondingto the inequality which Bradley and Wargentin originally derived fromobservation. We have already mentioned that Bailly succeeded in com-puting these inequalities by the aid of Clairaut ’s theory. Lagrange alsocalculated the value of the analogous inequality of the fourth satellite, buthe found it to he insensible. Delambre’s tables, in fact, make it only 11 s .

Considering next the inequalities dependent on the eccentricities, heobtained four equations of the centre for each satellite. One of thesedepended on the satellite’s own eccentricity, the other three were thereflected effects of the eccentricities of the disturbing satellites. He did notdetermine the maximum values of these equations, nor did he perceive theanalogy between those of the third satellite and the two equations of thecentre, which Wargentin deduced from observation. This is only one ofa number of instances in which this great genius failed to derive anysubstantial results from his brilliant researches.

Finally, he investigated the effects of the disturbing forces perpendicu-lar to the planes of the orbits, and obtained for each satellite four equationsof latitude similar to the four equations of the centre, to which his re-searches on the motion in longitude conducted him. He represented theposition of the orbit of each satellite by means of four planes passingthrough the centre of Jupiter . The first of these revolved upon the orbitof the planet with a constant inclination; the second revolved in a similarmanner upon the first; the third upon the second; and finally the fourth,

* “ Une analyse nouvelle, puissante, et dont les developpemens ont quelque chosed’efFrayant." Astronomie Theorique et Pratique, tome iii. p. 498.