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

18.1. Transformation theory 333
The central question of transformation theory. ABEL picked up the theme of trans-
formations of elliptic integrals from LEGENDRE. In ABEL’S words, the central problem
was posed with variations on multiple occasions, for instance in the following way:
“To find all the possible cases in which one can satisfy the differential equation
dy
�
(1 − y2 ) � 1 − c2 = a
1y2� dx
� (1 − x 2 ) (1 − c 2 x 2 )
by an algebraic equation between the variables x and y and supposing the moduli c
and c1 less than unity and the coefficient a either real or imaginary.” 6
Expressed in the notation of LEGENDRE’S differentials, the above question asks
for every possible way of transforming x algebraically into y in such a way that the
integral with modulus c1 transformed into the integral with modulus c.
18.1.1 ABEL’s response to JACOBI’s announcements
ABEL first published in the Astronomische Nachrichten in 1828; 7 in a lengthy paper, he
demonstrated how the theory which he had developed in his Recherches could answer
a question raised by JACOBI. In the paper — which is entitled Solution d’un problème
général concernant la transformation des fonctions elliptiques — , 8 ABEL began by describ-
ing key results concerning the inverse of elliptic integrals deduced in the Recherches.
With the notation
and the inversion
∆ (x) =
�
θ =
0
�
(1 − c 2 x 2 ) (1 − e 2 x 2 )
dx
and x = λ (θ) ,
∆ (x)
ABEL presented the highlights of the Recherches in two theorems which summarize
(16.3) and (16.8), respectively. First, he expressed the addition theorem for the elliptic
function of the first kind λ,
λ � θ ± θ ′� = λ (θ) ∆ (θ′ ) ± λ (θ ′ ) ∆ (θ)
1 − c2e2λ2 (θ) λ2 (θ ′ .
)
Second, ABEL described the conditions on the arguments which ensured that the func-
tion took identical values,
λ (θ) = λ � θ ′� if and only if θ ′ = (−1) m+m′
θ + mω + m ′ ω ′
6 “Trouver tous les cas possibles où l’on pourra satisfaire à l’équation différentielle:
dy
�
(1 − y2 ) � 1 − c2 = a
1y2� dx
� (1 − x 2 ) (1 − c 2 x 2 )
(18.1)
par une équation algébrique entre les variables x et y, en supposant les modules c et c1 moindre que
l’unité et le coeffcient a réel ou imaginaire.” (N. H. Abel, 1829a, 33).
7 (N. H. Abel, 1828d). The paper is dated 27 May 1828.
8 (ibid.).