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.

~ 122 —124.

Elliptic Integrals.

209

in which e may vanish; and F a rational combination of x and J/IB. For,
just as it was really indifferent in considering the previous class whether
the polynomial under the square root was of the first degree or of the
second - the integral could be reduced to a rational form in both
cases -, so also here the cases of the third and of the fourth degree
are equivalent: the integrals can be transformed into one another; they
no longer however in general become rational. This transformation is
effected by the substitution:
x = 7 + +-k c hi
x ==== k q- -,- = f-4k h
by which:
JR = a + lx + cx2 + dx3 + ex4 = f(x)
becomes
f(k + ]h) = (k) + -f- hf'(k) + - f"(c)+ 1. f2.3 f'(") + 1. 2 3 f'
or
f(7^ + - 1) = (2 +1 {( + in)4 f(k) (+ + mi)3 lf'(k) (Z + M)2 12f"(7k)
+ 2 3f"'(k) + f IV ()}
1.2.3 1.2.3.4
Now if we determine k so that f(k) = 0, i. e. that 7v be a root of the
equation R = 0, we have:
_ 1 1
g3 = (+,1' A,-' + B + Cv + D _ 2 1 _)
- (2+m)e /A -B C jD (2~4m)
A = lf'(), B = 3 8ilf' (7 ) + - 2f" (k),
C - 3m2 lf' (v) + m2l2f"(lv) + 1 2. 3f(k)
D = n mf'(k) + m2 /12f" (k) +.2-3 (7) + 1.23.4 f()
The quantities I and i remain arbitrary.
Under the square root there is now only a cubic polynomial,
and since x is rationally expressed by, fF(x, f/x)dx is transformed
intojO (-, J/Z) dz, as was stated. Denoting the roots of R = 0 by
a, 3,,, and making k coincide with a, the corresponding values of
S are: o, - - -- - i, -- mn. Therefore to one
3 P - c c m y —a d. — a
vanishing value of R corresponds a value of z that becomes infinite;
to the others correspond the roots of Z = 0.
The evaluation of the elliptic integral:*)
*) The geometric problem, to determine the length of the arc of an ellipse
or of a hyperbola between arbitrarily given terminal points, led to integrals of this
form. The Italian mathematician Fagnano (1682-1766) (Produzioni matematiche,
t. II, 1750) first found geometric relations among arcs of one of these curves by
HUARNACIC, Calculus. 14