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.

212

The integral of explicitly irrational functions. Bk. III. ch. III.

i. e. it is expressed, as already stated, by the three fundamental integrals:
r dx xdx d
VK'J 7R J (X- Q) VyR
these are called the elliptic integrals of the first, second and
third kind.
Employing the substitution which served to convert a polynomial
of the fourth degree under the square root into one of the third degree,
by transforming these integrals back again, we can also state our result
as follows:
Every elliptic integral f (x, I J)dx, in which:
B = a + bx + cx2 + dx' + ex4,
can be expressed by three fundamental integrals:
dx r dx r df
JO J (x-a) R' J (x - o) VR
and by algebraic and logarithmic functions.
In the integral of the second kind, a denotes a root of a R 0.
It can also be shown, by developing a formula of reduction analogous
to 1II., that:
Vx- andJ V2d
can be introduced instead of the last two integrals.
125. We now proceed still on the basis of these reductions to
establish the three normal integrals to the calculation of which
Legendre reduced the general elliptic integral.
Let the coefficients in R == a + bx -+- cx2 + dx3 + ex' be real,
then whether the four roots of R = 0 be real or complex, we can
always by a real linear substitution cause the odd powers in the
polynomial to vanish.
When the four roots are all real, let us name them so that
a > f > y > 6; when complex let oc be conjugate to /, and y to 6.
Putting
R = e(x - a)(x - f) (x-) (x- ) = e(x2-2- Xx+ )(x2-2 x+ ),
then:
a+ p= 2A, aPx =, y + == 2, y( == a.
From the substitution:
x -=
we have:
Az2 + AB + C ~ 2V A'2+ d - ' I C'
wx2 —2;Xx - A: x" 2x-[+ +
where: - 2 x + t (I + -2X(1+ + 5 ) --
where: