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

370 Chapter 19. The Paris memoir
which allowed him to rewrite (19.31) as
α =
=
n−1
∑ hqk + n − 1 − (hr − µ) + A
k=1
n−1
∑ hqk −
k=1
n
∑ hθ (yk) + n − 1 + A + µ. (19.32)
k=1
Therefore, ABEL turned to algebraically manipulating the “degrees” hθ (y k) along the
same lines as had been followed in describing γ above.
Obviously, from the inequality
and the formula
ABEL obtained
hθ (y) ≥ h (qmy m ) for each m = 0, . . . , n − 1,
h (qmy m ) = hqm + mhy,
hθ (y k) ≥ hqm + mhy k for k = 1, . . . , n.
Designating by ρτ the index of the maximal value of h (qmy m ) within the τ’th sequence
of roots, ABEL employed the same machinery which had served him before, although
this time in a slightly different notational dressing. Summing the excesses ε τ,k within
the τ’th sequence,
nτµτ−1
∑ ετ,k = Cτ,
k=1
ABEL found by the manipulations and number theoretic results
or less specifically (Cτ ≥ 0)
µ − α ≥ γ − A +
µ − α ≥ γ − A.
ε
∑ Cτ,
τ=1
However, the inequality in (19.33) was actually an equality,
µ − α = γ − A, (19.33)
as ABEL deduced by another tedious sequence of manipulations.
Specializing the relation expressed (19.33) to the other case (in which F0 (x) = 1)
led to the result that if r did not contain any factors independent of the indeterminate
quantities, then
µ − α = γ.
ABEL concluded these investigations by considering situations in which the coef-
ficients q0, . . . , qn−1 were subjected to some kinds of conditions. Again arguing very
“generally”, ABEL could claim that the result (19.33) would not generally be substan-
tially altered, although the constant A should reflect the additional conditions.
In the paper’s eighth section, ABEL applied the results obtained thus far to calcu-
late γ in an example for which n = 13.