5IR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 375 
but (3 = j-jrj 
. sin 2 e 
cos p sm <p = — 
and 1 — 2 sin 2 <p = cos 2 <p : wherefore. 
sin 4» sin 2 <p 
cos h + cos 2 <p ’ 
and, sin (2 <p — ip) = h sin ip. 
Wherefore if the quantities, 
h, k, k Y , k 2 , &c. 
represent a series of increasing- moduli derived from one another by these 
equations, 
v — 1 ~ \ l i — l ~K fa i — Lz k \ & c . 
h ~ 1 + h’ K \ ~ \ +k> k 2 — 1 + V ’ 
and further, if the amplitudes ip, <p, <p l5 <p 2 > &c. be deduced from the formulas, 
sin (2 (p — ip) = h sin ip, 
sin (2 <p x — <p) = k sin <p, 
sin (2 <p 2 — <p{) = k l sin <p x : &c. 
we shall have these transformations in which the successive moduli tend to 
the limit 1, 
F (h, iP) = (\ + k')F(k,<p), 
F (h, ip) = (1 + k') (1 -f ki) F (k 1} <Pi), 
F ( h , ip) = (1 + k') (1 -j- k{) (1 + k 2 ) F ( k 2 , , <p 2 ), &c. 
Example 2. Supposing^ = 3. 
By the formulas (2) and (5) we have these equations between the amplitudes 
ip and <p, 
sin ip = 
(\ — ) 
V sin 2 A 9 / 
1 — 
/r, 
sin 2 
COS ip = 
sm A 
wherefore, 
(l-^WA 2 ) 2 = /3^(l -^)' + (1-**) (l 
from which we get 
2 
2 k 2 sin 2 A 2 = + 1 - (3 
3 c 
MDCCCXXXI. 
