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

16.2. Inversion in the Recherches 303
ABEL’S derivation of the addition formulae. ABEL’S way of obtaining addition for-
mulae for elliptic functions resembles L. EULER’S (1707–1783) argument (see section
15.2.1) because both proceeded from a suggested formula. In ABEL’S case, he sought
to establish the identity
φ (α + β) =
φ (α) f (β) F (β) + φ (β) f (α) F (α)
1 + e 2 c 2 φ 2 (α) φ 2 (β)
and similar formulae for the auxiliary functions f and F,
f (α + β) = f (α) f (β) − c2 φ (α) φ (β) F (α) F (β)
1 + e 2 c 2 φ 2 (α) φ 2 (β)
(16.3)
and (16.4)
F (α + β) = F (α) F (β) + e2φ (α) φ (β) f (α) f (β)
1 + e2c2φ2 (α) φ2 . (16.5)
(β)
ABEL denoted the right hand side of (16.3) by r = r (α, β) and proceeded to dif-
ferentiate r with respect to α. The expression which he obtained after inserting the
values of f and F proved to be symmetric in α and β. Therefore, and because r itself
was symmetric in α and β, ABEL concluded that
∂r
∂α
∂r
= . (16.6)
∂β
This differential equation, ABEL claimed, 12 showed that r was a function of α + β,
r = ψ (α + β) . (16.7)
Upon inserting β = 0, ABEL immediately recognized ψ = φ, and the addition formula
for φ had been obtained.
To understand how ABEL concluded that the solution to the differential equation
(16.6) must be of the form (16.7), we can get a hint from one of his earlier papers,
published in the Journal. 13 In that paper, ABEL had established that the solution of the
equation 14
has the solution
� �
∂r
σ (y) =
∂x
��
r = ψ
�
σ (x) dx +
� �
∂r
σ (x)
∂y
�
σ (y) dy
where ψ was arbitrary. 15 To apply to the situation of the addition formulae, take σ = 1
to obtain (16.7).
12 The validity of this claim will be discussed below.
13 (N. H. Abel, 1826e).
14 ABEL’S use of d has been replaced by ∂; and ABEL wrote φ where I have substituted σ.
15 (ibid., 12–13).