204 
CALCULUS 
Finally, 
i 
_l c 
dz 
V ^ 
S 2 + 2 + 1 )(? + 1) 
. 1 + t 
, 2 dt 
~l-t’ 
(i -ty 
0 
1 /■ 
dt 
V2 J 
-1 
V(3 + i 2 )(l + i 2 ) 
To compute this integral in terms of F(k, </>), set (§ 2, (6)) 
— t 
Vl + * 2 
t = 
Vl -T 2 
1/V2 
V3 
C?T 
V(l-r ! )(l -|t ! ) 
5. Computation by Series. We have already seen how the func 
tions F(k, <f>) and E(k, <£) can be computed by infinite series ; Intro 
duction to the Calculus, pp. 414, 416. These series do not, however, 
converge rapidly when k is nearly unity. In this case, a transfor 
mation can be made (Landen’s Transformation, § 6) whereby either 
(a) k will be replaced by a smaller value, Aq, and thus the new 
series will become available for practical use ; or (b) k can be re 
placed by a still larger value, so near to unity that it may be set 
= 1 in the integral, and then the latter can be computed by means 
of the indefinite integral. 
6. Landen’s Transformation. We give the transformation with 
out motif* Starting with the integral 
<i> 
(!) F(k,-I>)= f-Jr = = 
J V1 — k 2 sin 2 <f> 
o 
we introduce a new variable of integration, by the equation 
(2) sin (21]/ — (f>) = k sin <f>, 
* The mathematicians of the eighteenth century were men of great resource 
fulness in formal work, and many of the leading results in the theory of the 
elliptic integrals and functions were deduced by inspiration rather than by 
reasoning. On the other hand, the modern theory of transformation of the 
elliptic transcendents is too complex to admit of a brief description.