No CrossRef data available.
Article contents
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
Published online by Cambridge University Press: 17 January 2013
Extract
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.
- Type
- Research Article
- Information
- Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 5 , Issue 2 , 1805 , pp. 253 - 293
- Copyright
- Copyright © Royal Society of Edinburgh 1805
References
page 255 note * Traité du Calcul Differential et du Calcul Integral, par Lacroix; vol. page 120.
page 256 note * Traité du Calcul Differentiel et du Calcul Integral, par Lacroix; vol. ii. page 120.
page 263 note * See a Memoir upon the Comparison of Elliptic Arcs, by Legendre, in the Memoirs of the Royal Academy, of Sciences for 1786. See also the Appendix to this Paper.
page 267 note * In a tract entitled “Animadversiones in Rectificationem Ellipsis,” which forms part of the second volume of his Opuscula.
page 267 note † Transactions of the Royal Society of Edinburgh, vol. iv.
page 272 note * This discovery was made by Mr Landen, who published it first in the Philosophical Transactions for 1775, and afterwards in his Mathematical Memoirs.
page 280 note * The properties of the ellipse here alluded to have been explained by Euler, and some of them have also been observed by Landen.
page 284 note * See Fig. page 279.
page 286 note * Hence, by the way, it appears, that instead of the semi-perimeters of two ellipses which we have used in the preceding paper, for expressing the coefficients of the development of (a 1 + b 2 — 2ab cos φ)n, we may substitute any two of an indefinite number of elliptic arches, and certain algebraic functions of the axes of these ellipses; therefore, the different infinite series, which may be used to express the coefficients A and B, are really innumerable.