MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 363 
:-(S) 
dz 
u ' 2 = t{ ( v + aU ) 
d. (V - aU) 
d z 
(V - aU) 
L(V + aU) 
d z 
from which it follows, that every double binomial factor either of V + a U, or 
of V — a U, is a simple binomial factor of 
U 2 ; and further, if V+ a U 
and V — a U have no common divisor, that every double binomial factor 
of (V -f a U) X (V — a U) = V 2 — a 2 U 2 , is a simple binomial factor of 
6. The differential of the equation (1) may now be readily demonstrated, 
supposing that (3 has the value investigated in § 4, and h, the value assigned to 
it in § 5. And first when p is an odd number, we obtain from the equation (2), 
, /3 . z P 
sin — , z — sin <p : 
and with these values the equation (1) will become 
V (R 2 - /3 3 * 3 P 2 ) (R 2 - /3 2 h* z* P 2 
l 
\/ 1 — s 2 . 1 — k 2 s 2 ’ 
and, on account of the formulas (C), 
Q. R' 
1. 
Now it is evident that R + (3 . z P and R — (3 . z P, have no common divisor : 
for, as R contains only the even powers of z, and z P only the odd powers, if 
1 + c z be a factor of R + j3.zP, 1-cj will necessarily be a factor of 
R — (3 . z P. Wherefore, according to what has been proved above, every 
double binomial factor of R 2 — (3 2 z 2 P 2 , that is, every factor of Q, will be a 
factor of the function in the numerator of the left side of the last equation. In 
the very same manner it is proved that every double binomial factor of 
R 2 — (3 2 h 2 z 2 P 2 , that is, every factor of R', will be a factor of the same function. 
