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HISTORY OF PHYSICAL ASTRONOMY.

the other hand, has obtained rT g T -g for the value of the mass by means ofthe perturbations produced by the planet in the motion of Saturn . It isto be hoped that the researches on the satellites, in which M. Otto Struveis known to be engaged at present, will lead to more satisfactory resultsrelative to this important point *.

The theory of the smaller planets still continues in a very imperfectstate. This circumstance is attributable to the magnitude of the eccen-tricities and inclinations, in consequence of which the disturbing functionconverges with such slowness as to render the usual methods of approxima-tion generally inapplicable. The only one of the ancient planets whichbears any analogy in this respect to those more recently discovered isMercury; but in this case the disturbed body is so near the sun, and atthe same time so remote from the larger planets of the system, that itsperturbations are very insignificant, and a small number of the terms ofthe disturbing function suffice for the calculation of all the inequalitiesthat are of sensible magnitude. The smaller planets, on the other hand,all revolve in the region comprised between the orbits of Mars and Jupiter ,and on this account their elliptic motions are very much deranged bythe powerful action of the latter planet. Attempts have frequently beenmade to investigate the perturbations of these bodies by the usual methodsof approximation, but their places when thus determined have been foundvery soon to present a marked discordance with those indicated by actualobservation. The perturbations of Vesta, Juno, and Ceres, have been com-puted with more or less success by Daussy, Santini, Damoiseau, and othergeometers; but those of Pallas, which revolves in an orbit, inclined at anangle of 34° to the ecliptic, have deterred even the most persevering analystsfrom undertaking their complete investigation. In recent times, attentionhas been principally directed towards the algebraic form of the disturbingfunction, with the view of devising modes of development, which shall bepracticable whatever be the magnitude of the eccentricities and inclina- ■tions. The researches of Cauchy , Liouville , Le Verrier , Hansen, andLubbock , in connexion with this subject, have resulted in various ingeniousprocesses by means of which it is to be hoped that this part of the planetarytheory will soon attain a degree of perfection, equal to that which is soconspicuous, when the question relates to the perturbations of the largerbodies of the system. Le Verrier has applied his method to the com-putation of a remarkable inequality in the mean motion of Pallas, occa-sioned by the disturbing action of Jupiter . This inequality depends uponthe near commensurability of the mean motions of Jupiter and Pallas.Eighteen times the mean motion of Jupiter , minus seven times the meanmotion of Pallas, forms a quantity which amounts to only yjgth of themean motion of the latter planet. Now, as this quantity appears in thedisturbing function under the symbols sine and cosine, the operation oftwo successive integrations will introduce its square into the denominatorsof the corresponding terms of the longitude. This circumstance may causethe term sto acquire a sensible magnitude, although in other respectsthey are very inconsiderable, being, according to the theory of planetaryperturbation, only of the eleventh order with respect to the eccentricitiesand inclinations. Le Verrier shewed it to he one of the advantages of hismethod, that the great inclination of the disturbed planet, so far from

* Dr. Lamont, of Munich , by observing the elongations of the satellites, lias obtained avalue for the mass of the planet considerably less than either of those mentioned in thetext. See Mem. Ast. Soe,, vol. xi.