374 MR, IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
by the first we pass from the greater modulus k to the less h ; and by the 
second, from the less modulus h to the greater k. 
In the first of the two cases we must derive the amplitude ^ from <p : and, 
for this purpose we immediately obtain from the formula (11), 
tan — cp) = k' tan <p. 
Wherefore, if 
k, h, h v h 2 , &c. 
represent a series of decreasing moduli, of which the complements are, 
k', 11, A/, h 2 , &c. 
the successive quantities being derived from one another by these formulas, 
7 _ 1 - k' 7 _ 1 - V 7 l - hi 0 
n ~ l + V’ n i ~ l + h'> ~ f TV’ &C ‘ : 
and, if we likewise deduce a series of amplitudes in this manner, 
tan (\p — <p) = k 1 tan <p 
tan (T — iP) = h! tan ^ 
tan (\p 2 — T) = V tan • T? &c. 
we shall have these successive transformations, by which the value of the given 
function F (k, <p) is made to approach indefinitely to the arc of a circle, 
F(k,p) =T F ( A >'M 
f (£,?>) = t • i T F ( /i i>+i) 
F (A,p) = T • T • T* • F (h 2 , 'W, &c- 
In the second case, when we would pass from the less modulus h to the 
greater Jc, the amplitude <p must be deduced from T For this purpose we 
have 
