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 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.

A particular class of dynamical problems has recently been treated geometrically by Sir Robert Stawell Ball, formerly astronomer royal of Ireland, now Lowndean Professor of Astronomy and Geometry at Cambridge. His method is given in a work entitled Theory of Screws, Dublin, 1876, and in subsequent articles. Modern geometry is here drawn upon, as was done also by Clifford in the related subject of Bi-quaternions. Arthur Buchheim of Manchester (1859–1888), showed that Grassmann's Ausdehnungslehre supplies all the necessary materials for a simple calculus of screws in elliptic space. Horace Lamb applied the theory of screws to the question of the steady motion of any solid in a fluid.

Advances in theoretical mechanics, bearing on the integration and the alteration in form of dynamical equations, were made since Lagrange by Poisson, William Rowan Hamilton, Jacobi, Madame Kowalevski, and others. Lagrange had established the "Lagrangian form" of the equations of motion. He had given a theory of the variation of the arbitrary constants which, however, turned out to be less fruitful in results than a theory advanced by Poisson.[99]