ELLIPTIC INTEGRALS 
205 
or its equivalent, 
(3) tan <p =— s ^' ^—, 
w r k + cos2if, 
or 
tan (cp — ip) — tan ip. 
1 + k 
From the first of the equations (3) we have: 
(4) d<t> = 2 
d4 = 2-- 1+fccos2 » <M. 
1 + 2 Zc cos 2 {]/ + k 2 
From (4) it appears that dip ¡d<p is always positive, and so, as cp 
increases from 0 to tr, the determination of ip with which we are 
concerned will increase from 0 to ir/2. 
Furthermore, from the first of equations (3), sin 2 </> can be com 
puted, and thus we find : 
1 4- k cos 2ip 
(5) 
Vl — k 2 sin 2 cp = 
Vl 2 k cos 2ip + k' 2 
From (4) and (5) we have: 
dcp _ 2 dip _ 2 
(6) 
dip 
Vl — k 2 sin 2 <p Vi + 2 k cos 2\p -\-k 2 1 + k Vl — k\ sin 2 ip 
(7) 
k\ = 
4 k 
]c = l - Vl - fcf 
1 +vr=rfei 
(1 + fc) 2 ’ 
On integrating equation (6) we find : 
4> \jj 
r dcp __ 2 r dip 
J Vl — k 2 sin 2 cp 1 + kj Vl — k\ si 
Sin 2 ip 
where the limits of integration, cp and ip, are connected by (2), or 
either of the forms (3). 
To sum up, then, we have : 
(8) 
F(k, cp) = 
1 -f- k 
F(ki, ip) 
k,= 
2Vk 
1 + k’ 
sin (2 ip — <p)= k sin <p. 
The new modulus, k 1} is greater than k, but less than 1. For, 
first, if 
2Vk 
1+k 
>k, 
then 
4 k 
(1 + k)' 
> k 2 and 4 > A: (1 -f- k) 2 .