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

19.2. The contents of ABEL’s Paris result and its proof 353
After these algebraic operations, ABEL now for the first time employed the calculus
in differentiating the equation (19.6) above. ABEL wrote the differentiation as
F ′ (x) dx + ∂F (x) = 0, (19.7)
where F ′ (x) represents the differential of F with respect to x and ∂F (x) represents
the differential of F with respect to all the indeterminates. This relationship was a
fundamental one, and ABEL immediately put it to use. He introduced the differential
dv =
µ
∑ f (xk, yk) dxk, k=1
where f was a rational function. This differential was the real object of concern in these
investigations. Through a sequence of deductions employing the theory of equations
(see example below), ABEL reasoned that dv was a rational function of the parameters
a1, . . . , aN. Therefore, its integral v would have to be expressible by algebraic and
logarithmic functions of these parameters 8 ,
v =
µ �
∑
k=1
f (x k, y k) dx k = algebraic and logarithmic terms.
This result is what I have termed Main Theorem I.
Theorem 16 (Main Theorem I) Under the present assumptions, the sum
µ �
∑
k=1
f (x k, y k) dx k
can be expressed by algebraic and logarithmic functions of the parameters a1, . . . , aN. ✷
An example of ABEL’S use of the theory of equations. In order to see that dv was
indeed a rational function of the parameters, ABEL first claimed that the simultaneous
equations (19.1) and (19.3) expressed y k as a rational function of x k, 9
y k = ρ (x k) .
Rearranging the equation (19.7) then produced
f (x, y) dx = −
f (x, ρ (x))
F ′ ∂F (x) = φ2 (x) .
(x)
Of this function φ2, ABEL observed that it was obviously rational in x and the param-
eters. Thus, dv could be rewritten as
dv =
µ
∑ φ2 (xk) (19.8)
k=1
8 The integral of any rational function was of course expressible by rational and logarithmic terms.
9 In order to see that ρ is rational as claimed, please observe that deg θ = deg χ − 1. See the proof of
lemma 4 in box 9.