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

20.2. Other ways of introducing elliptic functions in the nineteenth century 381
In this situation, ABEL found by employing Lagrange interpolation that the sum of inte-
grals of the third kind reduced to a logarithmic expression
µ
∑ Π (xn, a) = C −
n=1
a
2∆ (a)
f (a) + φ (a) ∆ (a)
log . (20.2)
f (a) − φ (a) ∆ (a)
ABEL extended this property of elliptic integrals of the third kind to analogous re-
sults for elliptic integrals of the first and second kind. For integrals of the second kind,
ABEL observed that ˜ω (x) = lima→∞ Π (x, a) whereas the logarithmic term vanished
under this limit process,
µ
∑
n=1
˜ω (xn) = C.
For integrals of the second kind, ABEL considered the expansion of (20.2) according to
increasing powers of 1 a
and compared coefficients of 1
a 2 to conclude
µ
∑
n=1
˜ω0 (xn) = C − p
where p was an algebraic function of x1, . . . , xµ.
Thus, ABEL used tools similar to those which he employed in the Paris memoir to
deduce results which also bear similarities with the Main Theorem I (theorem 16).
A new program of representability. In the second chapter, ABEL suggested a very
general question: He wanted to describe all integrals of algebraic differentials which
could be expressed by algebraic, logarithmic, and elliptic functions. Thus, if com-
pared with the Paris memoir, the elliptic functions were now accepted among the basic
functions to which other higher transcendentals could be reduced. However, when he
came to answer the question, he restricted himself to attack transformation problems
and other relations among elliptic integrals. The more specific contents of his inves-
tigations are considered outside the present scope, although its presentation would
probably reiterate the description of ABEL’S methods and tools which have been sug-
gested in the previous chapters.
ABEL did not manage to complete his papers before he died. Nevertheless, his
approach illustrated the fruitful influence which the results of Paris memoir could have
on the theory of elliptic functions. However, because of his early death, it was left
to ABEL’S contemporaries and competitors to outline the future development of the
theory of elliptic functions.
20.2 Other ways of introducing elliptic functions in the
nineteenth century
During the nineteenth century, the definitions of elliptic functions were turned upside
down a number of times. In chapter 16, it has been described, how ABEL introduced