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

344 Chapter 18. Tools in ABEL’s research on elliptic transcendentals
His investigations now turned toward solving the equation (18.12). ABEL did so by
observing that the process used above could be iterated producing a sequence of rela-
tions similar to (18.12). After n − 1 iterations, he found the relation
snβ 2 n−1 − 2rnβn−1βn − sn−1β 2 n = (−1) n−1 v,
in which δβn < δβn−1.
Because the sequence of degrees was decreasing, it would eventually produce δβm =
0, i.e. βm = 0, and the final relation would then become
smβ 2 m−1 = (−1)m−1 v.
Using this information, ABEL ascended the chain of β, β1, . . . , βm in the reverse order
each time finding expressions for βn−1 of the form
βn−1 = 2µnβn + βn+1.
By solving these relations for the first term of the chain β, ABEL found an expression
for β
β 1 as a finite continued fraction.
In order to answer the question of logarithmic integration of the original differen-
tial, ABEL next investigated the consequences for the radical √ R. He found from his
earlier results that by assuming m infinite, the expansion of √ R would be
√
R = t1 +
2µ +
1
1
2µ 1+ 1
2µ 2 +...
Here, ABEL noticed in a footnote that the equality of √ R and its continued fraction
should not be interpreted as a numerical equality except in those situations where the
continued fraction has a value.
Finally, ABEL translated an earlier assumption that one among the quantities s1, s2, . . .
should be independent of x into the property that the continued fraction for √ R
should be periodic. The assumption on s1, s2, . . . had been introduced to ensure the
solubility of the equations, and it thus amounted to a criterion for the possibility of in-
tegrating
ρ dx
√ R in logarithmic terms. ABEL summarized his investigations as a complete
criterion of logarithmic integrability stating that for polynomials ρ, the integration
� ρ dx
√R = log y + √ R
y − √ R
.
(18.13)
could be effected if and only if the expansion of √ R into continued fractions was
periodic. In the affirmative case, the function y was determined by the first period of
the continued fraction for √ R.