364 MR, IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
Wherefore the numerator of the left side of the last equation is divisible by 
the product Q X IV in the denominator ; and, as both the expressions have 
the same dimensions and the same absolute term, they are identical ; which 
verifies the equation. Wherefore the equation (1) is demonstrated when^ is 
an odd number. 
Secondly, when p is an even number, we have by equation (3), 
• . Z V 1 — P • 
5m ^ = V l-FF •K> sm ? = z: 
and the differential of equation (1) will become by substitution. 
( 1 — k 2 z~) R 2 J / z V 1 — z 2 . P \ 
dz ' Va/i -f-^Tr/ 
sj ^(1 — k° %•) R 2 — (3 2 z 2 (1 — z- ) P 2 ^ ^(1 — k 2 z 2 ) R 3 — /3 2 h 2 z 2 (l — z 2 ) P 2 ^ 
= — t-—: o 1 . n : and, on account of the formulas (D), 
Vl — z 2 .l - k 2 z 2 ’ v 
(i -F~z 2 ) R a A / z . P \ 
dz c * Wi_ k 2 z 2 . R/ 
Q.R' 
I 
V 1 — z 2 . 1 — k 2 z 2 
It will be proved, by the like reasoning as before, that the numerator of the 
left side of this equation is divisible by the product in the denominator. Now 
if we perform the differentiation indicated, we shall find, 
S = (1 - 2* 2 + k 2 z 4 ) PR+ * (1 - z 2 ) (1 - k 2 z 2 ) R 2 
(1 - FF) R 2 / z >/i - 
dz ' \V\ — Jc 2 z 2 * R/ — 
and it is evident that all the rational factors of the left side of this last formula, 
and consequently all the factors of Q X IV, will be factors of S. By substi- 
tution the differential equation (1) will now become 
_s ... _ 1 
Q x R' ~ ’ 
which is manifestly verified : for, as Q X IV divides S, and the two expressions 
