On the Reducibility of the General Elliptic Integral into Logarithms

In the following paper I have considered the question of the reduction of the general elliptic integral into logarithms. Abel has considered the problem in an algebraic manner and has given some sufficient conditions. Special cases were considered by Dolbnia, Goursat and Halphen and others. My aim in the following paper is to give a set of conditions which do not involve relations with the elliptic transcendentals, namely η, ω, etc. I give in the following paper a set of sufficient conditions for the reducibility, not involving the elliptic transcendentals in any way, which are believed to be new.