Section 1. General expression for an entire function

Translated by me as:

Let  f(x',x'',x'''...)  be any function that can be expressed as a finite number of terms of the form

                                                                 Ax' ^m₁ . x'' ^m₂ ......

where A is a quantity independent  of x', x'' etc. and m₁, m₂ etc are positive integers; it is clear that the operation designated by  f(x',x'',x'''...) is a special case of the first two steps above. We can therefore consider the entire functions according to their definition as resulting from a limited number of repetitions of this operation. Thus by appointing v', v'', v''' etc. as several functions of  x', x'', x''' ...., of the same form as f(x',x''...) the function f(v', v'' ...) will evidently be of the same form as f(x',x''...). Now f(v', v'' ...) is the general expression of functions resulting from the operation f(x',x''...) twice repeated. We therefore find the same result by repeating this as many times as you like. It follows that any entire function of several quantities x', x'', x''' ..... can be expressed by a sum of several terms of the form Ax' ^m₁ . x'' ^m₂ .....

Commentary:

Abel derives a general expression for any entire function.