The researches on functions mentioned thus far have been greatly extended. In 1858 Charles Hermite of Paris (born 1822), introduced in place of the variable of Jacobi a new variable connected with it by the equation , so that , and was led to consider the functions , , .[56] Henry Smith regarded a theta-function with the argument equal to zero, as a function of . This he called an omega-function, while the three functions , , , are his modular functions. Researches on theta-functions with respect to real and imaginary arguments have been made by Meissel of Kiel, J. Thomae of Jena, Alfred Enneper of Göttingen (1830–1885). A general formula for the product of two theta-functions was given in 1854 by H. Schröter of Breslau (1829–1892). These functions have been studied also by Cauchy, Königsberger of Heidelberg (born 1837), F. S. Richelot of Königsberg (1808–1875), Johann Georg Rosenhain of Königsberg (1816–1887), L. Schläfli of Bern (born 1818).[85]
Legendre's method of reducing an elliptic differential to its normal form has called forth many investigations, most important of which are those of Richelot and of Weierstrass of Berlin.
The algebraic transformations of elliptic functions involve a relation between the old modulus and the new one which Jacobi expressed by a differential equation of the third order, and also by an algebraic equation, called by him "modular equation." The notion of modular equations was familiar to Abel, but the development of this subject devolved upon later investigators. These equations have become of importance in the theory of algebraic equations, and have been studied by Sohnke, E. Mathieu, L. Königsberger, E. Betti of Pisa (died 1892), C. Hermite of Paris, Joubert of Angers, Francesco Brioschi of Milan. Schläfli, H. Schröter, M. Gudermann of Cleve, Gützlaff.
