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

366 Chapter 19. The Paris memoir
Investigation of the complementary case β ′ ≥ n. In order to study what happened
if β ′ ≥ n, ABEL wrote
µ1
m1
+ 1 = n + ε, ε ≥ 0
and obtained for the inverse fraction only two possibilities
m1
µ1
= 1 m1
or
n − 1 µ1
= 1
n .
In both cases, ABEL claimed, the integral � f (x, y) dx would be expressible in alge-
braic and logarithmic terms. His argument proceeded from claiming that the equation
χ (y) = 0 was linear in x,
χ (y) = P (y) + xQ (y) .
If this was the case, the integrand in � f (x, y) dx was quickly seen to be a rational
function, and the result was thus well known. However, as SYLOW has observed, 30
in the case left unnoticed above, the conclusion of linearity does not hold, and the
deduction thus suffers from this incompleteness.
Back on track: the case β ′ ≤ n − 1. Returning to the more complicated case, ABEL
noticed: “Thus, except for this case [β ′ ≥ n], the function f1 (x, y) always exists” 31 and
he went on to elaborate the consequences of the hypothesis β ′ ≤ n − 1. He began by
reducing the study of the equation
to the study of the individual terms
� n−β
∑
∑
′ −1
m=0 tmym χ ′ dx = C (19.26)
(y)
�
xkym dx
∑ χ ′ (y) .
His next and decisive step was to begin considering the htm + 1 coefficients in the
polynomial tm. He found that the function f1 (x, y) contained
n−β ′ −1
∑ (htm + 1) =
m=0
n−β ′ −1
∑ htm + n − β
m=0
′ =
n−2
∑ htm + n − 1 (19.27)
m=0
coefficients and chose to designate this number of coefficients by γ. Once this number
had been introduced, it became ABEL’S first objective to derive other general formu-
lae for it and to study certain particular cases. Once these investigations had been
concluded, ABEL again returned to (19.26), remarking that it was even valid in certain
cases not included in the deduction:
30 (Sylow in N. H. Abel, 1881, II, 298).
31 “Excepté ce cas donc, la fonction f1 (x, y) existe toujours” (N. H. Abel, [1826] 1841, 167).