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

288 Chapter 15. Elliptic integrals and functions: Chronology and topics
Rectification of the ellipse Consider the ellipse with major axis 2a and minor axis
2b given by the Cartesian equation
Obviously, from this equation
x 2
a
b
y2
+ = 1.
2 2
x = a cos θ and y = b sin θ
which means
dx
dy
= −a sin θ and = b cos θ.
dθ dθ
To compute the arc length, we find
�
s (θ) =
� �
dx
dθ
� �
= b
� 2
+
� �2 �
dy �
dθ = a
dθ
2 sin2 θ + b2 cos2 θ dθ
1 − k 2 sin 2 θ dθ with k 2 = b2 − a 2
This is precisely the form of A.-M. LEGENDRE’S (1752–1833) second kind of elliptic
integrals with the modulus k (denoted F (θ, k) in table 15.2).
In order to reduce the integral to the form � R(x) dx
, we substitute z = sin θ and get
and thus
dz = cos θ dθ =
√ P4(x)
b 2
�
1 − sin 2 θ dθ ⇒ dθ =
� �
1 − k2 sin2 �
θ dθ =
√ 1 − k2z2 �
√ dz =
1 − z2 Box 6: Rectification of the ellipse
.
1
√ 1 − z 2 dz
1 − k 2 z 2
� (1 − k 2 z 2 ) (1 − z 2 ) dz.
Figure 15.1: EULER’S rectification of an ellipse by infinite series (L. Euler, 1732a, 2)