206 
CALCULUS 
This last inequality is true, since 0 < k < 1; and now, starting 
with it, we can retrace our steps. In a similar manner it is shown 
that k x < 1. 
If k is nearly = 1, a few repetitions of the transformation (7) will 
lead to a function F(k n , <f> n ), whose modulus k n is so nearly unity 
that it may be replaced by 1 and the integral thus evaluated. Since 
2 
1 + fc 
we have : 
On setting k n = 1, we find : 
K 
Tor a detailed study of a numerical case, cf. Byerly, Integral 
Calculus, 2d ed., chapter on Elliptic Integrals. 
Reducing the Modulus. The transformation (7) can be applied in 
the opposite sense, and thus the given integral is referred to one 
with smaller modulus. The formulas now become: 
l+Vl + frf 
Here, k x and ^ are given, and k and cf> are computed from the 
second line of (9). The student may find it convenient to rewrite 
(9), interchanging <f> with ^ and k with k v A numerical example is 
worked in detail in Byerly’s book, l.c. 
After one or two applications of the transformation (9), it may be 
well to finish the computation by using the series. 
Landen’s 
Integrals of the Second Kind, 
transformation can be applied to these, too, and thus the computa 
tion carried through ; cf. Byerly, l.c. An excellent treatment of 
this subject, including also the rectification of the hyperbola and the 
lemniscate, and the complanation of the central quadrics, is found 
in Schlomilch, Compendium der hdheren Analysis, voi. 2, 2d ed.