An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 125.

Reduction of elliptic integrals to normal forms.

217

is only of the third degree, that it can be reduced to the three
fundamental forms:
r 't 4'tdt - dt
accordingly Legendre's normal integrals expressed in y are:
VJ01(i - ) ci - l^~ (r-y — y)/ ( -— kr y —Y- y )'
or as Le gendre restricting himself primarily to the investigation of
real values of the integral, by substituting
y = sin p, dy =cos q d p
wrote them:
(p cp (p
P) f, Z (pd)-J 7 ( \" sin2 p)cCp d _
o U_ 0
A p = J/1 -- 71' sin2 ll.
1
The coefficient n - -- in the third integral is called the Parameter
Q
of the third normal integral.*)
Note. From the central equation of the ellipse
2 = 1, a > b27
introducing ( the eccentric anomaly reckoned from the axis minor,
we have:
x = a sin p, y == bcos (p
f X2 / d ya y2 - dJ p /a2 cos2 (2p + b2 sin 2 =- a (/i - /2sin2 p,
a2 b2
7k2= 2 < 1.
Thus the length of the elliptic arc that belongs to the values 0 and
(p depends on the calculation of the integral:
a(>=a c cs CdP 12 inT d
aE((p) --- a A 0- 1j —k sin. 2-a F(p) —kZ(jp)}.
o 0 o
From the central equation of the hyperbola - 2 1 in
which a means the length of the real semiaxis, putting x = a sec p,
we find y=- b tan Tp,
J/dx2 + dy2 - dq sec2 cp /'2 - 2 sin2 (p.
*) A more compendious account of the transformation to the normal form,
due to Weierstrass, in which the coefficients of a fractional linear substitution are
I
determined so that the values y = - 1, + -- shall correspond to the four roots
of R = 0, is communicated by Schellbach: Die Lehre von den elliptischen Integralen und den Thetafunctionen; and Konigsberger: Vorlesungen fiber die Theorie
der elliptischen Integrale.