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.

218

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

To obtain integrals of the previous form, however, we introduce the distance from the centre to the focus, c2 - a2 + b2, and put
cy = b2 tan (p, then:
x- a sec p J/1 - 7L sin2 g, c21c2 -a2,
thus:
/dx2~- - - -- y/i+ - dl/f+( +
+ y2fat a dy Vb + y, (a, + bl)
SY~=S~i~1t x~ b2 be x
1p
b2 f dqp - 7/7 -A== - ~ = // 1 -- I sin2 (p.
c J cos2p A V1 sing.
0
Therefore the integral Y(gp) by which the length of the hyperbolic
arc from -- 0 is measured, is directly equal to that third normal
integral TT((p) in which the parameter m) = - 1. But when n =- 1,
this third integral can be reduced to the second and first, it is the
case of the vanishing value 9 of the linear function coinciding with
a root of Y = 0. From the identity:
d(Ay. tan Y() Ap d k 'sinp d p
d' p - cos2 p A
(1 - 2) -d7'2\ -t d1 712 - inpq
cos'' A A Acp
we have:
Y(gp)- A cp t8an ) p - ^- a-() + a Z(~)v
C
or: Y(g)) - c A tan T +- -2- F(q)) - cE((p).
The first term has a simple geometric meaning. It is equal to
the length alolg the tangent to the curve at the point belonging to
gp, measured from that point to the foot of the perpendicular let fall
from the centre.*)
126. Elliptic integrals of the three kinds, or the three normal
integrals of L e g e nd r e are calculated by means of expansions obtained
by converting the function to be integrated into a series of powers and
forming the integral of this infinite series. But this method requires
some preliminary general investigations.
*) Legendre: Traite des fonctions elliptiques, p.' 16.