Considerations such as these have been used for determining the series of values of the independent variable, and the irrational functions thereof in the theory of Abelian integrals, but the theory seems to be worthy of further investigation.
Riemann's results are contained in the memoirs on " Abelian Integrals," &c. (Crelle, t.
Observe that the radical, square root of a quartic function, is connected with the theory of elliptic functions, and the radical, square root of a sextic function, with that of the first kind of Abelian functions, but that the next kind of Abelian functions does not depend on the radical, square root of an octic function.
Valuable also are his papers on Abelian transcendents, and his investigations in the theory of numbers, in which latter department he mainly supplements the labours of K.
Liv., 1857) in which he applied the theory developed in his doctor's dissertation to the Abelian functions.