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 He concluded that they consisted of an aggregate of unconnected particles.

The problem of three bodies has been treated in various ways since the time of Lagrange, but no decided advance towards a more complete algebraic solution has been made, and the problem stands substantially where it was left by him. He had made a reduction in the differential equations to the seventh order. This was elegantly accomplished in a different way by Jacobi in 1843. R, Radau (Comptes Rendus, LXVII., 1868, p. 841) and Allégret (Journal de Mathématiques, 1875, p. 277) showed that the reduction can be performed on the equations in their original form. Noteworthy transformations and discussions of the problem have been given by J. L. F. Bertrand, by Émile Bour (1831–1866) of the Polytechnic School in Paris, by Mathieu, Hesse, J. A. Serret. H. Bruns of Leipzig has shown that no advance in the problem of three or of $$\scriptstyle{n}$$ bodies may be expected by algebraic integrals, and that we must look to the modern theory of functions for a complete solution (Acta Math., XI., p. 43).[98]

Among valuable text-books on mathematical astronomy rank the following works: Manual of Spherical and Practical Astronomy by Chauvenet (1863), Practical and Spherical Astronomy by Robert Main of Cambridge, Theoretical Astronomy by James C. Watson of Ann Arbor (1868), Traité élémentaire de Mécanique Céleste of H. Resal of the Polytechnic School in Paris, Cours d' Astronomie de l'École Polytechnique by Faye, Traité de Mécanique Céleste by Tisserand, Lehrbuch der Bahnbestimmung by T. Oppolzer, Mathematische Theorien der Planetenbewegung by O. Dziobek, translated into English by M. W. Harrington and W. J. Hussey.

During the present century we have come to recognise the advantages frequently arising from a geometrical treatment of mechanical problems. To Poinsot, Chasles, and Möbius we