356 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
any odd multiple (2n-\-\)co is substituted for u in the expression (B), the 
value of y is always + 1 or — 1> according as (2 n + 1) u holds an odd or an 
even rank in the series of the odd multiples of a. 
On the other hand, Avhen u is zero, or equal to 2 na any even multiple of a, 
we shall have y = 0, one of the factors of the numerator necessarily vanishing; 
for in a sequence of the even multiples of co, of which the number is p, there 
must be one equal to 2 pco, or to a multiple of 2 pco; and therefore when 
u — 2 n co, one of the factors must be the sine of an amplitude equal to 1 80° or 
to a multiple of 180°. 
Further, let co — z be substituted for u in the expression (B), a being less 
than a ; then, 
sin A (co — s) sin A (3 co — s) .... sin A (2 p co — co — z) 
y — y 
Now, in the numerator, the partial products, of the first and last factors, of the 
second and last but one, and so on, are as follows : 
sin A (co — 2) sin A (2p co — co — z) = sin A (a — z) sin A (co -j- z), 
sin A (3 co — z) sin A (2 p 00 — 3 a — z) = sin A (3 u — z) sin A (3 a + z), &c. 
to which we must add the single factor sin A (p a — z), when p is an odd num- 
ber. All the partial products, it will be observed, have the same value whe- 
ther z be positive or negative; and they are all greatest, when £ = 0, as will 
readily appear from what is proved in § 2. Wherefore y has the same value 
and the same sign, when u is at equal distances from the limits 0 and 2 a ; and 
it attains its greatest magnitude, equal to 1, when u = oo. And, if we substi- 
tute (2m+ 1) co — z for u, this substitution will not change the foregoing 
factors, but only their order, and the sign of their product, which sign, while 
u is contained between the limits 2 nu and 2w»-)-2«, will be + or — , accord- 
ing as (2 11 + 1 ) co holds an odd or an even rank in the series of the odd mul- 
tiples of u. 
We may now conclude, from what has been proved, that y , in the expression 
(B), represents the sine of an arc '<p, which increases from zero with the elliptic 
function u, and coincides with the successive terms of the series, 
7T 7T 7T . . . 
0, y, 2 y, 3 y, &c. ad infinitum, 
