358 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
0 = 
sin 5 A 2 . sin 5 A 4 . sin 5 A 6 . . . sin 2 X p _ 4 
sin 2 X x . sin 2 A 3 . sin 2 a 6 . . . sin Ap _ 2’ 
we shall have, 
being an odd number, 
sin 4> = (3 z 
sin 8 A 2 
1 - 
sin 2 A. 
1 - 
sin 2 Ap _ 1 
1 — k 2 z* sin 2 A 2 " 1 — Jc 2 z 2 sin 2 A 4 ’ ’ * 1 — /c 2 2 2 sin 2 Ap — 1 * 
( 2 ) 
When p is an even number, if we leave out the first factor in the numerator 
of the expression of y or sin there will remain an odd number of factors, 
that which occupies the middle place, being sin A (u -j- p co ) : and any factor, 
as sin A (u + 2 nu), between the first and the middle one, will have another, 
viz. sinA(«< + 2 poo — 2 nu), corresponding to it after the middle one; and 
the product of this pair of factors will be obtained as before, viz. 
sin A (u + 2 n a) sin A {u + 2 p a — 2 n a) = 
sin 5 A 2w - sin 2 <p 
1 — Jc 2 sin 3 A„ sin 2 0 
2 n r 
With regard to the middle factor, we shall have, in the formulas of § 2, 
sin a = sin A (p a) = sin 90°, sin b = sin A u = sin <p, sin s = sin A (u + p &>) ; 
and 
sin A {u + p u) = 
cos 
V 1 — k 2 sin 2 <p 
Wherefore, by proceeding as before, we shall have, 
p being an even number, 
sin 4 1 — 
/3 2 \/ 1 — z 3 
1 - 
si- 
sin 2 A 2 
1 - 
sin 2 A 4 
1 - 
2; 2 
sin 2 Ap _ 2 
Vl -F 2 2 1 — /r 2 ; 2 sin 2 A 2 1 — Jc 1 z 1 sin 2 A 4 1 — k~ z~ sin 2 Ap _ 2 
sin 2 A 2 . sin 2 A 4 . sin 2 A 6 . . . sin 2 Ap _ 2 
sin 2 A, . sin 2 A 3 . sin 2 A 5 . . . sin 2 Ap _ 1’ 
(3) 
In both the formulas (2) and (3), it is obvious that (3 is the quotient of the 
product of the sines of all the even amplitudes, X 2 , X 4 , &c. between the limits 
0 and 180°, divided by the product of the sines of all the odd amplitudes, 
/.j, X 3 , &c. contained between the same limits. The general expression of (3, 
common to the two cases, is therefore as follows, 
0 
sin Aj . sin A 4 . sin A 6 . . . sin X^p — 2 
sin A! . sin A3 . sin A s . . . sin X 2p _ j’ 
(4) 
