MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 359 
In the formula (2) let P and R stand for the products of the binomials in 
the numerator and denominator ; then, 
sin 4/ = 
/3sP 
R ; 
and. 
o , R 2 — /3 2 * 2 P s 
cos 2 y = . 
XL'' 
The numerator of this expression is a rational function of z 2 , and it will vanish 
whenever cos 2 4' = 0, or sin 2 %// = 1, that is, when z 2 is equal to sin 2 A 2rt + i, 
2 n -f 1 being any odd number less than 2 p. Suppose that 2 n + 1 is any odd 
number less than p, the numerator of the value of cos 2 ^ will be divisible by 
( 1 r— ^ ), and also by A and as these binomials are 
\ Sin 2 A2« + 1/ \ Sin" \2p — 2n— 1 / 
( 2 2 X 2 
1 — sin o A2 +1 j ’ and, p being itself 
an odd number, to the double divisors there must be added the single one 
\ • « # 
1 — s - n , — j = 1 — z 2 . The numerator is therefore divisible by the product, 
C 1 — z 2 ) • ~ sin- A,) • ( 1 ~ sin- A 3 ) ( 1 “ sin 2 Ap _ 2 ) : 
and, as the two expressions have the same absolute term and the same dimen- 
sions, they must be identical. Wherefore we have, 
p being an odd number, 
l — 
cos 
sm 2 A 
I - 
sm s Ao 
\L i ; — 
Y 1 — k- z 2 sin- A 2 1 — fc 2 z 2 sin 2 A 4 
sin 2 A» _ 2 
1 — k 2 z 2 sin 2 A: 
i> • 
( 5 ) 
In like manner, if P and R represent the rational binomial products in the 
numerator and denominator of the formula (3), we shall have 
and 
• , 3 z */ 1 — z 2 P 
sl ^- 7r-tv x B 
2 , _ (1 -£ 2 z 2 )R 2 -/3 2 z 2 (l — z°~) P z 
T t 1 7,2 ~‘2\ T? 2 
(1 — lc 2 z 2 ) R 2 
Proceeding as before, it will appear that the numerator of this expression is 
3 A 
MDCCCXXXI. 
