Section 1. General expression of an algebraic function: Part 1

Translated by me as:

We will now find the general form of algebraic functions. Denote by f(x', x''...) any rational function whatsoever. It is clear that every algebraic function can be made using the operations designated by f(x', x''...)  combined by the operation    ^m√ r where m is a prime integer. So, if p', p''... are rational functions of x', x''... then

                                                                p₁ = f(x', x''...^n'√ p', ^n''√ p''...)

is the general form of the algebraic function of x', x''... in which the    ^m√ r affects only rational functions. The functions of the form  p₁ will be called first order algebraic functions. We designate by p₁', p₁'' multiple quantities of the same form as  p₁ . The expression

                                                p₂ = f(x', x''...^n'√ p', ^n''√ p''... ^n₁'√ p₁', ^n₁''√ p₁'' ...)

is the general form of the algebraic function of x', x''... in which the    ^m√ r affects only rational functions and algebraic functions of the first order. Functions of the form p₂ will be called second order algebraic functions. In the same manner the expression

                                  p₃ = f(x', x''...^n'√ p', ^n''√ p''... ^n₁'√ p₁', ^n₁''√ p₁'' ...^n₂'√ p₂', ^n₂''√ p₂'' ...)

in which p₂', p₂'' are algebraic functions of the second order, is a general expression of an algebraic function in  x', x''... in which the    ^m√ r affects only rational functions and first and second order algebraic functions.

Commentary: 

Abel develops the concept of the order of an algebraic function and then expresses the general form of an algebraic function up to the third order. We can see that in general these algebraic functions will be multi-valued.