Section 1. General expression of an algebraic function: Part 7

Translated by me as:

In the forgoing we have seen, a rational function of several individual quantities can always be reduced to the form

                                                                                    s
                                                                                    t

where s and t are entire functions of the same variable quantities. It is concluded from this that v can always be expressed as follows
                                                                          φ(r', r''...ⁿ√ p)
                                                                v =  ————————
                                                                         τ(r', r''...ⁿ√ p)

where φ and τ are entire functions of r', r''... and ⁿ√ p. Based on the above we have found that  any entire function of several quantities s, r', r'' ... can be expressed by the form

                                                                   t₀ + t₁ s + t₂ s ² + ... t_m s ^m

t_{0 ,} t_{1 ...} t_m  being entire functions of r', r'', r'''... without s. We can therefore let

                                                             t₀ + t₁ p ^{1 / n} + t₂ p ^{2 / n} + ... t_m p ^{m / n}                T

                                                  v =  ————————————————————  =   ——
                                                             v₀ + v₁ p ^{1 / n} + v₂ p ^{2 / n} + ... v_m' p ^{m' / n                  V}


where  t_{0 ,} t_{1 ...} t_m   and  v_{0 ,} v_{1 ...} v_m'  are entire functions of r', r'', r''', etc.

Commentary:

Abel expresses  algebraic function in a way which emphasizes it's dependence on one of it's higher order quantities  ⁿ√ p. This expression is very general and perhaps the only restriction Abel places on it is that the quantity ⁿ√ p is one of it's highest order terms, which means that it should be at least tied in terms of having the greatest number of nested radicals. The m introduced here is different from the m previously introduced to represent the degree of the algebraic function