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of twenty-five, he published his Fundamenta Nova Theoriœ Functionum Ellipticarum, which contains in condensed form the main results in elliptic functions. This work at once secured for him a wide reputation. He then made a closer study of theta-functions and lectured to his pupils on a new theory of elliptic functions based on the theta-functions. He developed a theory of transformation which led him to a multitude of formulæ containing $\scriptstyle {q}$, a transcendental function of the modulus, defined by the equation $\scriptstyle {q=e^{-\pi k^{'}/k}}$. He was also led by it to consider the two new functions $\scriptstyle {H}$ and $\scriptstyle {\Theta }$, which taken each separately with two different arguments are the four (single) theta-functions designated by the $\scriptstyle {\Theta _{1}}$, $\scriptstyle {\Theta _{2}}$, $\scriptstyle {\Theta _{3}}$, $\scriptstyle {\Theta _{4}}$.^[56] In a short but very important memoir of 1832, he shows that for the hyperelliptic integral of any class the direct functions to which Abel's theorem has reference are not functions of a single variable, such as the elliptic $\scriptstyle {sn}$, $\scriptstyle {cn}$, $\scriptstyle {dn}$, but functions of $\scriptstyle {p}$ variables.^[56] Thus in the case $\scriptstyle {p=2}$, which Jacobi especially considers, it is shown that Abel's theorem has reference to two functions $\scriptstyle {\lambda (u,v)}$, $\scriptstyle {\lambda _{1}(u,v)}$, each of two variables, and gives in effect an addition-theorem for the expression of the functions $\scriptstyle {\lambda (u+u^{\prime },v+v^{\prime })}$, $\scriptstyle {\lambda _{1}(u+u^{\prime },v+v^{\prime })}$, algebraically in terms of the functions $\scriptstyle {\lambda (u,v)}$, $\scriptstyle {\lambda _{1}(u,v)}$, $\scriptstyle {\lambda (u^{\prime },v^{\prime })}$, $\scriptstyle {\lambda _{1}(u^{\prime },v^{\prime })}$. By the memoirs of Abel and Jacobi it may be considered that the notion of the Abelian function of $\scriptstyle {p}$ variables was established and the addition-theorem for these functions given. Recent studies touching Abelian functions have been made by Weierstrass, E. Picard, Madame Kowalevski, and Poincaré. Jacobi's work on differential equations, determinants, dynamics, and the theory of numbers is mentioned elsewhere.

In 1842 Jacobi visited Italy for a few months to recuperate his health. At this time the Prussian government gave him a pension, and he moved to Berlin, where the last years of his life were spent.