In the cases then we have just enumerated, there is no real difficulty in computing the perturbations in longitude and parallax, by means of the differential equations of pp. 258. 268. and the value of R. The series of terms expressing that value of R will not extend, by reason of the decreasing values of A, B, C, beyond a certain limit: nor will the terms of the several series that express the values of A, B, C, D. But we must now consider cases before adverted to, namely, those in which radius of the orbit of the disturbed planet radius of the orbit of the disturbing should not differ so much from unity, as it differed in the preceding cases. Let us suppose an extreme case, and that two of the newly discovered planets, Juno and Ceres, whose mean distances are 2.667163, 2.767406, are the mutually disturbing bodies. In such a case, it is clear, that the terms A, B, C, D, &c. would neither decrease rapidly, nor would the terms of the series that severally represent their values. The solution of the problem would, without some new device, become impracticable. But the preceding case may be thought too unimportant to shew the necessity (practically speaking) of some new method of computation. It is, otherwise, however, with the Earth's perturbations by the action of Venus, which are required to be known in the construction of the Solar Tables. Now, in such a case a = .723323, and the terms of the series of which we have spoken, in the above value of a, decrease so slowly as to be, at the least, extremely incommodious. Some new artifice is requisite for their exact summation. The above remarks apply to the cases of the mutual perturbations of Mars and the Earth; of Jupiter and Saturn, and of Jupiter's Satellites acting on each other. Euler, investigating the mutual perturbations of Jupiter and Saturn, perceived the failure of the ordinary methods for computing the coefficients A, B, &c. and first invented a new : method. Clairaut*, on the subject of the perturbations of the Earth by the action of Venus, invented a different method, but for the same end as Euler's. Other methods have been subsequently invented, and, Science being progressive, the last invented are better than the preceding. But of such methods there are none that are simple and obvious: and the least simple, and least obvious, but, by many degrees, the most commodious, is the one which will be described in the next Chapter. Besides its immediate and practical importance, it will illustrate the manner by which the refined and abstruse formulæ (as they are called) of Analysis, may be made subservient to the ends of Physical Science. * Lorsque l'orbite de la planéte troublante est considerablement plus grande ou plus petite que celle de la planète troublée, les series qu'expriment la distance de deux planetes et ses puissances, se presentent tout naturellement sous une forme assez convergente; mais dans les cas ou les rayons des deux orbites ont un rapport qui ne permet pas de negliger les puissances élevées, les mêmes series decroissent si peu, qu'il faut avoir recours a des artifices particuliers pour determiner avec precision les termes dont on a besoin. Telle est la question de l'action de Venus sur la terre, qui nous reste a traiter dans ce Memoire. Telle est aussi celle de l'action de Jupiter sur Saturne que M. Euler a consideré dans la pièce que l'Academie couronna en 1748; c'est cet habile Geomètre qui a trouvé le premier la reduction des series de l'espece dont nous avons besoin maintenant'. Acad. des Sciences. 1754. p. 545. CHAP. XVIII. On the Method of determining the Coefficients of the Development of r (r-2rr' cos. wr2)-2s when the Fraction does not differ much from 1. Application of the Formula to the Mutual Perturbations of the Earth and Venus. THE investigation in the present Chapter, will consist of two parts: one, the deduction of the coefficients C, D, E, and from the two first A and B; the other, the numerical computation of A and B this latter point will be first considered. and, multiplying each side of the equation by the differential of w and integrating dw √ √ - 2. cos. w+) = 4+ B sin. + C.sin. 2 + &c. let designate the semi-circumference of a circle, the radius of which is 1, and suppose the integral of the above equation to be taken within the values of w = 0, and ∞ = π, then dw √(1 2 p'. cos. w +p'2) and the difficulty now (see p. 284.), under somewhat of a different shape, is to find the preceding integral. Now, the differential expression on the right-hand side of the equation is such, that if we assume and, accordingly, if we continue to make like assumptions, viz. P and designating respectively, the nth terms of the series p', p", "", &c. u', u', u", &c. The advantage of this last form will be obvious, if we consider, that, p' must be less than the square of p, less therefore than the square of a fraction, for such p is always supposed to be: similarly, p" must be less than the square of p", p", less than p"; and so on the series, therefore, p', p", p"", &c. must be a rapidly decreasing one; so that, a term P will be soon arrived at, so small as to enable us to neglect P2 V2. Now if we may neglect P2 V2, the difficulty of finding the inwill be reduced to that of find dx tegral of π however, between the values of √ = 0, and V = 1, is equal to (3.14159): the only ques 2 tion, therefore, that remains to be decided, is concerning the values of x, corresponding to the values of V = 0, and V=1: and, in order to determine it, we must examine the values of u', u", &c. consequently, u= 0, both when x = 0, and when x = 1, and it is at its intermediate and maximum value, when |