362 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
in which expression R and P denote the same functions of % as before, and the 
values of the new symbols h and R' are as follows, 
h = k p sin 4 X 1 . sin 4 X 3 . sin 4 X 5 . . . sin 4 X p _ 2 , 
R' = (l — k 2 z 2 sin 2 X x ) (1 — k 2 z 2 sin 2 X 3 ) . . . (1 — k 2 z 2 sin 2 X p _ 2 ). 
We thus have 
JR 2 - (3 2 s 2 P 2 = JT^7 2 . Q 
^R 2 - j3 2 h 2 z l P 2 = J\ - k 2 z 2 . R'. 
Secondly, when p is an even number, R, P, Q will stand for the rational 
binomial products in the denominators and numerators of the equations (3) 
and (6) : thus 
. /3 2 v' 1 — z 3 P , 1 Q 
SU1 ^ — 1 - * "R J C0S ^ ~ V 1 - /r z 8 * R : 
consequently 
(1 - k 2 z 2 ) R 2 = /3 2 z 2 (1 - ^ 2 ) P 2 + Q2. 
And if in this equation we substitute yz in place of z, we shall obtain this 
result, 
(1 - k 2 z 2 ) R 2 = (3 2 z 2 (1 - z 2 ) P 2 + R' 2 
R and P representing 1 the same functions of z as before, and the new symbols 
h and R' standing for these values, 
h — k p . sin 4 . sin 4 X 3 . sin 4 X 5 . . . sin 4 X p _ x 
R' = (1 — k 2 z 2 sin 2 Xj) (1 — k 2 z 2 sin 2 X 3 ) . . . (1 — k 2 z 2 sin 2 X p _ x . 
From what has been proved we now have 
I - k 2 z 2 ) R 2 - (3 2 z 2 (1 - z 2 )Y 2 — Q, 
^/(l - k 2 z 2 ) R 2 - ,3 2 h 2 z 2 (1 - z 2 ) P 2 = R' 
To these formulas we must add the following principle of analysis, on which 
the demonstration we have in view mainly turns. Let V and U denote rational 
functions of z : we shall have this identical equation, 
