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

334 Chapter 18. Tools in ABEL’s research on elliptic transcendentals
in which the semi-periods were determined by
� �
ω 1
= θ and
2 c
ω′
2
= θ
� �
1
.
e
ABEL expressed the central properties of this theorem:
“This theorem is generally valid no matter whether the quantities e and c are
real or imaginary. In the paper cited above [Recherches], I have proved it in the
case where e 2 is negative and c 2 is positive. [. . . ] The quantities ω, ω ′ always have
an imaginary ratio. Otherwise, they have the same role in the theory of elliptic
functions as the number π has in the theory of circular functions.” 9
Now, in order to address the transformation problem, ABEL observed that the
method of indeterminate coefficients could be applied. This method amounted to ap-
proaching the problem by introducing two power series with indeterminate coeffi-
cients and using the defining equations to obtain relations among the terms. How-
ever, as ABEL critically remarked, this method would lead to extremely cumbersome
calculations, and ABEL proposed a simpler and more direct one. Below, this method
is briefly described.
Rational transformations. With the notation and basic results set up, ABEL turned to
a question which he proposed and ascribed great importance for the theory of elliptic
functions. He was interested in finding all the possible ways in which the differential
equation
dy
��1 − c2 1y2� � 1 − e2 = ±a
1y2� dx
� (1 − c 2 x 2 ) (1 − e 2 x 2 )
(18.2)
could be satisfied in which y was an algebraic function of x. In the paper, ABEL limited
his considerations to rational functions y = ψ (x) because the general question “at first
seems too difficult”. 10
ABEL’S first result in this situation was an algebraic one, not so different from re-
sults obtained in his paper on Abelian equations. 11 By a string of manipulations, ABEL
found that if the equation (18.2) was satisfied, the roots of the equation ψ (x) = y had
the remarkable property of being related in a very specific way: if λ (θ) represented
one of the roots, any other root of the equation would be representable as λ (θ + α)
where α was a constant, i.e.
y = ψ (λ (θ)) = ψ (λ (θ + α)) .
9 “Ce théorème a lieu généralement, quelles que soient les quantités e et c, réelles ou imagainaires.
Je l’ai démontré pour le cas où e 2 est négatif et c 2 positif dans le mémoire cité plus haut [. . . ]. Les
quantités ω, ω ′ sont toujours dans un rapport imaginaire. Elles jouent d’ailleurs dans la théorie des
fonctions elliptiques le même rôle que le nombre π dans celle des fonctions circulaires.” (N. H. Abel,
1828d, 366).
10 (ibid., 365).
11 See chapter 7.