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 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).93
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
owe the most important developments made in geometrical mechanics. Louis Poinsot (1777–1859), a graduate of the
Polytechnic School in Paris, and for many years member of the superior council of public instruction, published in 1804 his Éléments de Statique. This work is remarkable not only as being the earliest introduction to synthetic mechanics, but also as containing for the first time the idea of couples, which was applied by Poinsot in a publication of 1834 to the theory of rotation. A clear conception of the nature of rotary motion was conveyed by Poinsot's elegant geometrical representation by means of an ellipsoid rolling on a certain fixed
plane. This construction was extended by Sylvester so as
to measure the rate of rotation of the ellipsoid on the plane.