XI. A New Method of expressing the Coefficients of the Development of the Algebraic Formula (a2 + b2 − 2ab cos φ)n, by means of the Perimeters of two Ellipses, when n denotes the Half of any Odd Number; together with an Appendix, containing the Investigation of a Formula for the Rectification of any Arch of an Ellipse

Abstract

In calculating the effect of the mutual action of two planets upon each other, it has been found necessary to develop the algebraic formula (a2 + b2 — 2ab cos φ)n into a series of this form, A + B cos φ + C cos 2φ + D cos 3φ + &c. Here a and b denote the distances of the planets from the sun; φ denotes the angle of commutation; and the values of n, more immediately the subject of consideration, are —, and —.The determination of the coefficients A, B, C, &c. in these cases, appears to have been considered as a matter of difficulty by the mathematicians who first applied to the solution of the problem; for they found, that although it was only necessary to compute the first two coefficients A and B, the rest being easily derived from them, yet it did not appear that they could be expressed in finite terms, nor even by means of circular arches, or by logarithms. Recourse was therefore had to other methods, and chiefly to the method of infinite series; but as the series which most readily occurred to them, converged in some cases so slowly as to be in a manner useless, no small degree of analytical address has been found necessary, either to render it more convergent, or to find the sum of a competent number of its terms, with a moderate degree of labour.