Sur la méthode horistique de Gyldén. Offprint from: Acta Mathematica, vol. 29

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First edition, very rare offprint, of a later work by Poincaré on the three-body problem, following his great 290-page work 'Sur le probleme de trois corps et les equations de dynamiques' (1890). The present paper is a critique of Gyldén's 1892 work 'Nouvelles recherches sur les séries employées dans les théories des planetes.' The usual methods used to solve problems in celestial mechanics, especially the three-body problem, result in expressions involving 'small denominators', i.e., terms involving powers of 1/p which become infinite when p vanishes. Gyldén tried to show that if when using the traditional methods a more exact calculation is made, then these terms will not arise, and instead there will be what he called 'horistic' terms, in which 1/p is replaced by the square root of 1/(p^2 + r^2), where r is a very small but non-vanishing quantity. If Gyldén was right, this would imply that the series could be proved to converge. Poincaré showed in the present paper that Gyldén had made mistakes in the approximations at the beginning of his analysis, and in his understanding of what convergence meant. Poincaré also showed the falsity of Gyldén's conclusion that high-order terms in the perturbation function could never cause libration. Poincaré concluded with remarks that showed his frustration: "Several of Gyldén's results are clearly correct, but they could have been reached by a much quicker method; a great number are clearly false; most of them are given in such a way which is too obscure to decide whether they are true or false" (quoted in Barrow-Green, Poincaré and the three-body problem, p. 166). 4to, pp. 235-271. Original printed wrappers (browned and slightly chipped at edges, a few small tears, some marginal damp-staining in text). N° de ref. del artículo ABE-1575319728675

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