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.

Liv., 1857) in which he applied the theory developed in his doctor's dissertation to the **Abelian** functions.

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.

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.