MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 357 
at the same time that u attains the values, 
0, u, 2 u, 3 co, &c. ad infinitum, 
or, when the amplitude of u becomes equal to the several known arcs, 
0, X 2 , X 3 , &c. ad infinitum : 
and further, that there is but one value of y, or of sin between the two con- 
secutive terms m X and (m + 1) X for any given value of u between the 
limits mu and (m + 1) u, or for any given amplitude between the arcs X m and 
-(- 1 - 
4. In what has been proved, p may be either an odd or an even number ; 
but we must now distinguish between the two cases, in like manner as it is 
necessary to do when we investigate the sine of a multiple of a circular arc. 
Representing the amplitude of u by <p, we shall have, u — K (<p), and sin <p 
= sin A u. When p is odd, there will be an even number of factors after the 
first in the numerator of the expression of y or sin ^ ; and any one of these, 
as sin A ( u + 2 n u), will have another, namely, sin A (u + 2 p u — 2 nu) 
= sin A (2 n u — u ), answering to it ; and the product of this pair of factors, 
viz. sin A (u -j- 2 n a) X sin A (2 nu — u ), will be found by the formula (A) of 
§ 2, observing that sin a — sin A (2 nu) = sin X 2 „, sin b = sin A u — sin sin s 
= sin A (2 nu u), sin a = sin (2 nu — u) : 
thus we have, 
s in 2 — sin 2 <p 
sm A (u + 2 n u) sin A (u 4- 2 p u — 2 nu) = ■ : - w , . 
v ’ v 1 ’ 1— Fsin 2 sin 3 <p 
2 n T 
Wherefore, by taking in all the factors and writing z for sin (p, we shall obtain, 
sin 3 X 2 — z 2 sin 2 A 4 — z 2 sin 2 X p — i — z 2 
1 — k 2 z 2 sin 2 A 2 ’ 1 — k 2 z~ sin 2 A 4 " ‘ 1 — k 2 z 2 sin 2 A ;) _ i " 
The expression of y, viz. 
y = sin Xj . sin X 3 . sin X 5 . . . sin X 2p _ i, 
may be written in this form, 
y = sin 2 X x . sin 2 X 3 . sin 2 X 5 . . . . sin 2 Xp _ i, 
omitting the factor sin X p = 1 : wherefore, if we assume. 
