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 says that the manuscript was so badly written as to be illegible, and that Abel was asked to hand in a better copy, which he neglected to do. The memoir remained in Cauchy's hands. It was not published until 1841. By a singular mishap, the manuscript was lost before the proof-sheets were read.

In its form, the contents of the memoir belongs to the integral calculus. Abelian integrals depend upon an irrational function $$\scriptstyle{y}$$ which is connected with $$\scriptstyle{x}$$ by an algebraic equation $\scriptstyle{F(x,y)=0}$. Abel's theorem asserts that a sum of such integrals can be expressed by a definite number $$\scriptstyle{p}$$ of similar integrals, where $$\scriptstyle{p}$$ depends merely on the properties of the equation $\scriptstyle{F(x,y)=0}$. It was shown later that $$\scriptstyle{p}$$ is the deficiency of the curve $\scriptstyle{F(x,y)=0}$. The addition theorems of elliptic integrals are deducible from Abel's theorem. The hyperelliptic integrals introduced by Abel, and proved by him to possess multiple periodicity, are special cases of Abelian integrals whenever $$\scriptstyle{p=}$$ or $\scriptstyle{>3}$. The reduction of Abelian to elliptic integrals has been studied mainly by Jacobi, Hermite, Königsberger, Brioschi, Goursat, E. Picard, and O. Bolza of the University of Chicago.

Two editions of Abel's works have been published: the first by Holmboe in 1839, and the second by Sylow and Lie in 1881.

Abel's theorem was pronounced by Jacobi the greatest discovery of our century on the integral calculus. The aged Legendre, who greatly admired Abel's genius, called it "monumentum aere perennius. During the few years of work allotted to the young Norwegian, he penetrated new fields of research, the development of which has kept mathematicians busy for over half a century.

Some of the discoveries of Abel and Jacobi were anticipated by Gauss. In the Disquisitiones Arithmeticœ he observed