RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

19.2. The contents of ABEL’s Paris result and its proof 361
details. If v is a constant, say v = 0, the basic objects of the inquiry satisfy
�
0 = ∑
f (x k, y k) dx k.
The end product of the present section of the paper is the Abelian Theorem (Main Theo-
rem II) states something about exactly such sums of related integrals.
ABEL next expanded the relevant function, whose derivative was rational in x by
(19.8), according to decreasing powers of x,
∑ f1 (x, y)
χ ′ (y)
log θ (y) = R log x +
∞
∑ Akx k=0
µ0−k
, (19.18)
where R was “a function of x independent of a, a ′ , a ′′ , etc.,” 21 A0, A1, . . . were independent
of x, and µ0 designated an integer. 22
If expression (19.18) were to express a constant (independent of the indeterminates
a, a ′ , a ′′ , . . . ), ABEL observed that since these quantities occurred in A0, A1, . . . , the sec-
ond term corresponding to these had to vanish. And since he was only concerned
with the coefficient of 1 x , his conclusion was that µ0 < −1. He expressed this using a
newly introduced notational advance in the following sentence:
“This done, in designating by the symbol hR the highest exponent of x in the
development of any function R of this quantity following decreasing powers, it is
evident that µ0 will be equal to the largest integer contained in [less than or equal
to] the numbers
h f1 (x, y ′ )
χ ′ y ′
, h f1 (x, y ′′ )
χ ′ y ′′
�
f1
, . . . h
x, y (n)�
χ ′ y (n)
.
It is necessary that all these numbers must be less than the unit taken negatively.” 23
21 “R étant une fonction de x indépendante de a, a ′ , a ′′ , etc.” (ibid., 161).
22 The version printed in the Savants étrangers read at this point
⎧
⎨
R log x =
⎩
+Aµ0
A0x µ0 + A1x µ0−1 + . . .
+ Aµ0+1
x
+ Aµ0+2
x 2
+ . . .
In the collected works (Sylow in N. H. Abel, 1881, II, 296), SYLOW commented: “C’est évidemment
une faute d’écriture, ou d’Abel ou de Libri.” After the original manuscript has been recovered, it
has become evident that the misprint is indeed due to LIBRI.
23 “Cela posé, en désignant par le symbole hR le plus haut exposant de x dans le développement d’une
fonction quelconque R de cette quantité, suivant les puissances descendantes, il est clair que µ0 sera
égal au nombre entier le plus grand contenu dans les nombres:
h f1 (x, y ′ )
χ ′ y ′ , h f1 (x, y ′′ )
χ ′ y ′′ �
f1 x, y
, . . . h
(n)�
χ ′ y (n)
;
il faut donc que tous ces nombres soient inférieurs à l’unité prise négativement.” (N. H. Abel, [1826]
1841, 161).