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

18.2. Integration in logarithmic terms 343
and because NR1 = R, ABEL had found that the highest power in R had to be an even
number. ABEL wrote δN = n − m and δR1 = n + m and generalized the study of the
equation (18.10) to the equation
in which v was an entire function with δv < δN+δR 1
2
ABEL then wrote
p 2 1 N − q2 R1 = v (18.11)
= n.
R1 = Nt + t ′ with t = E R1
N and δt′ < 0.
As a consequence of the assumptions, ABEL found that δt = 2m and he wrote t in the
form
t = t 2 1 + t′ 1
in which δt ′ 1 < m. With these conventions, ABEL had remodelled the equation (18.11)
into
After rewriting the equation as
ABEL observed that
and he applied lemma 3 to obtain
v = p 2 1N − q2 � Nt + t ′� =
�
= p 2 1 − q2t 2 �
1 N − q 2 � t ′ 1N + t′� .
�
p 2 1 − q2 �
t N − q 2 t ′
� �2 p1
= t
q
2 v
1 +
Nq2 + t′ t′
1 +
N ,
�
v
δ
Nq2 + t′ �
t′
1 + < m = δt1,
N
E
� �
p1
= ±Et1 = ±t1.
q
This meant, that ABEL had found a relation between p1 and q of the form
p1 = t1q + β in which δβ < δq.
Through a sequence of similar, very explicit manipulations, ABEL transformed the
equation (18.11) into the form
in which
s1β 2 − 2r1ββ1 − sβ 2 1 = v (18.12)
δr1 = 1
2 δR = n, δβ1 < δβ, δs < n, and δs1 < n.