360 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
2 2 
divisible by the double divisor ^1 — ^ , 2 n + 1 being any odd num- 
ber less than p ; and in this case when p is an even number, all the divisors 
are double. Wherefore the product 
/ 1 _ y ( i fi-Y (\ A—) 2 (\ — y 
^ sm-x l J * V sin-Ag/ ' y sin 2 A 5 / * ' * * y sin 2 X p _ i / 
will divide the numerator of the value of cos 2 ^ ; and it will be identical to it, 
because both the expressions have the same dimensions. Thus we obtain, 
p being an even number, 
l - 
sin 2 Aj 
1 — 
s 2 
sin ' 2 A, 
1 - 
sin 2 A p — i 
cos <p y' i — /f 3 s 2 * 1 — ^ sin 2 a 2 1 — k 2 s 2 sin 2 X p - 2 
From the equations (2) and (5) we deduce. 
tan 4* 
_ /3. 
1 - 
sin 2 A 0 
1 - 
s 2 
sin 2 A. 
sin 2 x p — 1 
4/1 
ST 
cv2 
but it will readily appear that 
1 - 
sin 2 ip 
sin 2 A, 
“2 n 
sin 2 A, 
tan 2 p 
tan 2 A„ , 
1 - 
sin 2 a 3 
1 - 
sin 2 Xp - 2 
1 - 
sin 2 <p 
1 - 
tan 2 <p 
tan- A 2 n -)- 1 
sm-x 2n+1 
wherefore we obtain, 
p being an odd number, 
tan 2 p 
tan 4* = (3 tan <p X 
l - 
tan 2 X„ 
1 - 
tan 2 <p 
tan 2 A 4 
1 - 
tan 2 <p 
tan 2 Xp — 1 
1 - 
tan 2 <p 
tan 2 Aj 
1 — 
tan 2 <p 
tan 2 A, 
1 — 
tan 2 <p 
tan 2 X p - 2 
And in a similar manner we deduce from the equations (3) and (6), 
p being an even number. 
tan 4 1 = — 
tan 2 <p 
1 - 
1 - 
tan 2 <p 
tan 2 Ag 
1 - 
tan 2 <p 
tan 2 A, 
1 — 
tan 2 <p 
tan 2 Xp — 2 
tan 2 A 
1 — 
tan 2 f 
tan 2 Ao 
1 - 
tan 2 p 
tan 2 At 
1 — 
tan 2 <p 
tan 2 Xp - 1 
( 6 ) 
(7) 
( 8 ) 
The formulas (2), (5), (7), in which p is an odd number, are those used in 
