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 name "calculus of variations," and did much to improve this science along the lines marked out by Lagrange.

Another subject engaging the attention of Lagrange at Turin was the propagation of sound. In his papers on this subject in the Miscellanea Taurinensia, the young mathematician appears as the critic of Newton, and the arbiter between Euler and D'Alembert. By considering only the particles which are in a straight line, he reduced the problem to the same partial differential equation that represents the motions of vibrating strings. The general integral of this was found by D'Alembert to contain two arbitrary functions, and the question now came to be discussed whether an arbitrary function may be discontinuous. D'Alembert maintained the negative against Euler, Daniel Bernoulli, and finally Lagrange,—arguing that in order to determine the position of a point of the chord at a time $\scriptstyle{t}$, the initial position of the chord must be continuous. Lagrange settled the question in the affirmative.

By constant application during nine years, Lagrange, at the age of twenty-six, stood at the summit of European fame. But his intense studies had seriously weakened a constitution never robust, and though his physicians induced him to take rest and exercise, his nervous system never fully recovered its tone, and he was thenceforth subject to fits of melancholy.

In 1764 the French Academy proposed as the subject of a prize the theory of the libration of the moon. It demanded an explanation, on the principle of universal gravitation, why the moon always turns, with but slight variations, the same phase to the earth. Lagrange secured the prize. This success encouraged the Academy to propose as a prize the theory of the four satellites of Jupiter,—a problem of six bodies, more difficult than the one of three bodies previously solved by Clairaut, D'Alembert, and Euler. Lagrange overcame the difficulties, but the shortness of time did not permit him to