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.

216

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

square root is to be taken negatively; if s increase from + - to -+ co,
y increases from 0 to + 1, the square root is positive. We have:
dz 1 dy2 2
=z^ _i_ o --- 7]2 ~ 2
1/Z ~y/a+:2 + 2 / - y2) (1 -- y') 2 -+ 2 2
7) Z =- y2(1 - a2z2) (1 + f2s2) remains positive, for z from
1
_ x to t, and from + — to + co. Let us put:
C / y -- 2 =d ydy, thus we have:
l _ 2 (1 -- d 2)
dg 1 dy, a2
8) Z -- 2(l - a2z2) (1 - p2z2) is positive, for z from - - to, and from + - to + - Let us put:
3 f i- P2(32 U22 d, 2z=
13g =J g1 I2 y P2 pd, - 2 2 - z
a ory2)~~~a y 2 — '2
If z increase from - to -, y increases from 0 to + 1, the
1 1
square root is negative; ifz increase from -- to + -, y increases from
- 1 to 0, the square root is positive. We have:
d ___ 2 d 2 __ - 2
YZ py (I - y2)(I - y2) a
This concludes the proof of our assertion for all cases. To all values
of z for which 1/Z remains real, correspond values y2 < 1.
But if we observe that all the substitutions we have employed
are included in the form:
-V 2 + Y1y2Y
~ + S y2
we obtain the result: The elliptic integral f/-(2) Iz, disregarding
constant factors, can always be brought to the form:
_ (+ - ' ) dy where Y (1 - 2) (1 - y2)
E- stEy2 I dy
in which, provided Z has real coefficients, Z -- the modulus of
the elliptic integral - signifies a real positive proper fraction.
Putting y2 = t in order to complete the reduction of this integral
to the normal forms of Legendre, we have:
f' 4(,6:Y )dI f F(t) cit
iw ( +4) l, n wi2 t l
J f v^ + s/2 V /Y J /t( - t) (t — k7t)
It was proved (~ 124) of this integral, in which the polynomial