HISTORY OF PHYSICAL ASTRONOMY.

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turbed planets *. As the terms to be taken into account rise in degreethe labour of computing the coefficients increases with frightful rapidity.With the view of facilitating the researches of geometers, Burchardtcalculated all the terms of the perturbing function to the sixth powersof the eccentricities f. It still remained to execute a similar calcula-

tion with respect to the inclinations. This laborious task was per-formed by Binet, who carried the developement to the seventh powersof the eccentricities and inclinations. This has been found to be amplysufficient for computing the perturbations of the older planets, butit is by no means so, when the question refers to the small planets lyingbetween the orbits of Mars and Jupiter , or when the perturbations ofcomets are considered. In both these cases the eccentricities and incli-nations are so considerable that the terms of the disturbing function con-verge with extreme slowness; and on this account, it is necessary to cal-culate a much greater number of them than in the theory of the olderplanets, in order to attain an equal degree of accuracy. Geometers appearto have abandoned all hopes of determining the coefficients of the higherterms by the ordinary process of algebraic developement, the operationrapidly assuming so complicated a form as to become totally unmanageable.Other methods have accordingly been devised for effecting this object.One of these is founded on the principle of mechanical quadraturesM. Hansen has determined the coefficients of the disturbing function bythis method in a memoir on the theory of Jupiter and Saturn which ob-tained for him the prize of the Academy of Sciences of Berlin.

Another method consists in giving particular values to the disturbingfunction, and then, by means of the equations formed between them andthe corresponding values of the series, eliminating as many coefficients asmay he deemed desirable. This method has been practised with completesuccess by M. Le Verrier in computing the perturbations of Uranus bySaturn §.

* We have mentioned in the preceding note that R contains a series of terms of theform P cos ^ (in — i'n') t -j- . s — «' + Q ) Now the lowest terms which involve any

given values of i V are of the order i — i' relative to the eccentricities and inclinations.The next lowest terms having the same arguments would be of the order i — i' + 2; andthe order of the succeeding term swould increase by 2 at each step. For example, ifi = 5 and V = 2, then the lowest terms of the form

P cos ^ (5n ~ 2w') t + 5 s <- 2 i -\-

would be of the third order; the next lowest of the fifth order, and so on. Hence weperceive the reason why the terms upon which the long inequality of Jupiter and Saturn depends, are at least of the third order. Similarly in the long inequality of t’he Earthand Venus , which arises from the smallness of 13 n — 8 n f , the terms are at least of thefifth order.

f Memoires de l’lnstitut, 1808.

} This method, the germ of which may be traced to D’Alembert , forms the subject oftwo papers by Poisson , which appear in the volumes of the Connaissance des Temps for1825 and 1836. Pontecoulant has explained it in his Tkeorie Analytique du Systems duMonde, and has also given an example of its application in computing the great inequalityof Jupiter and Saturn . See the work just cited, tome iii. liv. vi. chap. vi. vii. xxi.

§ Rechercb.es sur le Mouvement de la Planete Herschel, Connaissance des Temps,1849. Le Verrier does not give the details of his investigation of the perturbations ofUranus by this method, his object being merely to shew the accordance of its results withthose he obtained by the ordinary method of algebraic developement. In a memoir alreadypublished by him ( Developpemens sur differents points de la Thtorie des Perturbations

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