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 's theory of the variation of the arbitrary constants and the method of integration thereby afforded marked the first onward step since Lagrange. Then came the researches of Sir William Rowan Hamilton. His discovery that the integration of the dynamic differential equations is connected with the integration of a certain partial differential equation of the first order and second degree, grew out of an attempt to deduce, by the undulatory theory, results in geometrical optics previously based on the conceptions of the emission theory. The Philosophical Transactions of 1833 and 1834 contain Hamilton's papers, in which appear the first applications to mechanics of the principle of varying action and the characteristic function, established by him some years previously. The object which Hamilton proposed to himself is indicated by the title of his first paper, viz. the discovery of a function by means of which all integral equations can be actually represented. The new form obtained by him for the equation of motion is a result of no less importance than that which was the professed object of the memoir. Hamilton's method of integration was freed by Jacobi of an unnecessary complication, and was then applied by him to the determination of a geodetic line on the general ellipsoid. With aid of elliptic coordinates Jacobi integrated the partial differential equation and expressed the equation of the geodetic in form of a relation between two Abelian integrals. Jacobi applied to differential equations of dynamics the theory of the ultimate multiplier. The differential equations of dynamics are only one of the classes of differential equations considered by Jacobi. Dynamic investigations along the lines of Lagrange, Hamilton, and Jacobi were made by Liouville, A. Desboves, Serret, J. C. F. Sturm, Ostrogradsky, J. Bertrand, Donkin, Brioschi, leading up to the development of the theory of a system of canonical integrals.