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ings depending on them to be represented in the most general form, in analytical language. D'Alembert applied it in 1744 in a treatise on the equilibrium and motion of fluids, in 1746 to a treatise on the general causes of winds, which obtained a prize from the Berlin Academy. In both these treatises, as also in one of 1747, discussing the famous problem of vibrating chords, he was led to partial differential equations. He was a leader among the pioneers in the study of such equations. To the equation $\scriptstyle {{\frac {\partial ^{2}y}{\partial t^{2}}}=a^{2}{\frac {\partial ^{2}}{\partial x^{2}}}}$, arising in the problem of vibrating chords, he gave as the general solution,

$\scriptstyle {y=f(x+at)+\phi (x-at)}$,

and showed that there is only one arbitrary function, if $\scriptstyle {y}$ be supposed to vanish for $\scriptstyle {x=0}$ and $\scriptstyle {x=l}$. Daniel Bernoulli, starting with a particular integral given by Brook Taylor, showed that this differential equation is satisfied by the trigonometric series

$\scriptstyle {y=\alpha \sin {\frac {\pi x}{l}}\cdot \cos {\frac {\pi t}{l}}+\beta \sin {\frac {2\pi x}{l}}\cdot \cos {\frac {2\pi t}{l}}+\cdots }$,

and claimed this expression to be the most general solution. Euler denied its generality, on the ground that, if true, the doubtful conclusion would follow that the above series represents any arbitrary function of a variable. These doubts were dispelled by Fourier. Lagrange proceeded to find the sum of the above series, but D'Alembert rightly objected to his process, on the ground that it involved divergent series.^[46]

A most beautiful result reached by D'Alembert, with aid of his principle, was the complete solution of the problem of the precession of the equinoxes, which had baffled the talents of the best minds. He sent to the French Academy in 1747, on the same day with Clairaut, a solution of the problem of three bodies. This had become a question of universal inter-