MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 355 
For the sake of abridging, let a — ■— K ; so that X 1} X 2 , X 3 , &c. will be the 
respective amplitudes of a, 2 a, 3 a, &c. ; and \ = amplitude of p a = amp. of 
K = 90° ; and X 2p = amp. of 2 p a = amp. of 2 K = 180°. From the nature of 
the integral, it follows that when u receives an addition equal to 2pa or 2 K, 
the amplitude of u will be increased by 1 80°. 
To the indeterminate quantity u let there be added the several even multi- 
ples of a less than 2p a ; and let us consider the sines of the amplitudes of the 
functions so formed, viz. 
sin A u, sin A (u + 2 a), sin A (u + 4 a), sin A (u + 2 p a — 2 a) : 
in this series, if we substitute in place of u, the successive quantities u-\-2a, 
m + 4 a, u + 6 a, &c., the same sines will constantly recur in periodical order, 
abstracting from the change of sign when an amplitude becomes greater than 
1 80°, or than a multiple of 1 80°. Thus, if we put u + 2 a in place of u, the 
second term of the foregoing series will stand first, and the last term will be 
sin A (u + 2 p a) = — sin A ( u ). In like manner, if u -J- 4 a be substituted for 
u, the third term of the series will stand first, and the two last terms will be 
— sin A u, — sin A (u + 2 a ) ; and so on. 
Let us now put 
y = sin A a X sin A (3 a) X sin A (5 a) X sin A (2 p a — • a) 
or, which is the same thing, 
y = sin \ X sin X 3 X sin X 5 . . . . X sin X 2p _ 1 ; 
and further, let us assume, 
sin Au x sin A (u + 2 a) x sin A (u + 4 co) . . . . x sin A (u + 2 p co — 2 m) 
y— (ii) 
In this expression, if we substitute for u , the several odd multiples of a in suc- 
cession, viz. 
a, 3 a, 5 a, 7 &C. 
it follows, from what has been said, that the products in the numerator will 
always be the same, and equal to the denominator, but that their signs will 
change alternately as the successive quantities are substituted. Thus, when 
