‎"JACOBI, C.G.J. - ABELIAN TRANSCENDENTAL FUNCTIONS.‎
‎Considerationes generales de transcendentibus Abelianis. - [ABELIAN TRANSCENDENTAL FUNCTIONS]‎

‎Berlin, G. Reimer, 1832. 4to. Without wrappers. Extracted from ""Journal für die reine und angewandte Mathematik. Hrsg. von A.L. Crelle"", 9. Bd., Heft 4, pp. 313-403. Jacobi's paper pp. 394-403. With titlepage to vol. 9.‎

Reference : 41658

‎First printing of this important paper, bringing the problem with the Abelian integral to a solution. ""The further step was made by Jacobi in the short but very importent memoir ""Considerationes generales..."" (the paper offered): viz. he here shows for the hyper-elliptical integrals of any class (but the conclusion may be stated generally) that the direct functions to which Abel's theorem has reference are not functions of a single variable, such as the elliptic sn, cn, or dn, but functions of ""p"" variables...(Arthur Cayley in his ""Presidential Address to the British Association 1883).""Carl Jacobi in reviewing Adrien Legendre's third supplement to his ""Traite des fonctions elliptiques et des integrales Euleurienne"", suggests renaming Legendre's ""hyperelliptical transcendental functions"" as ""Abelian transcendental functions"" after Niels Henrik Abel. acobi, whiose work on hyperelliptic integrals helps lead to the extensive nineteenth-century development of the theory of abelian functions of n variables, suggests, in partial analogy to doubly periodic functions, that hyperelliptic integrals can be inverted to hyperelliptic functions through a generalization of elliptic theta functions."" (Parkinson in ""Breakthroughs (1832"").‎