MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 369 
X- X 2 X 2 
1 -f- 7 TT~ 1 “p 7 o~T~ 1 "I - 7 5~7 
_ o x x tan-A^ _ tan- A 4 . . . tan-^-i ^ 
y ' x 2 x 2 
1 tan 2 Aj 1 tan 2 A 3 * tan 2 A P - \ 
In this equation the values of y and x are between 0 and + 1, which limits 
they both attain at the same time. If we make x = + 1, and attend to the 
value of (3, we shall find?/ = + 1. Let y = sinT, x = sin<r: then the inte- 
gral of the differential equation will be 
F (h' } r) = (3F (k', <r) 
i + 
sin t = (3 sin <r X 
snr <r 
tan 2 A 2 
1 + 
sin- <7 
tan 2 A 4 
1 + 
tan 2 An 
sin 2 o- 
sin 2 <r 
sin- <7 
( 12 ) 
" r tan 2 Aj tan 2 A 3 * tan 2 A p — 2 J 
the amplitudes r and <7 increasing together from zero, and becoming equal to 
one another at 90°, and at every multiple of 90°. 
A property of considerable importance in this theory, results from the com- 
parison of the equations (1) and (12). Recalling the notations before used, 
viz. K = F (k, y') and H = F (h, —J, we obtain from what has already been 
said in § 1, 
p X H = (3 X K: 
and if we put similarly K' = F (k\~^j and H' = F and observe that in 
the equations (12), r and a are equal to 90° at the same time, we shall have, 
H' = /3 K'. 
By combining the two equations, we readily obtain, first. 
H _ l K „ H 
H' p ' K' ’ ^ “ p ' K ; 
(13) 
and secondly, 
(3(3' = p; 
K' 
K 
1 H' 
p * H 5 
n , K' 
P=r-w 
(14) 
For any number p, the first of the formulas (13) determines h , and the second 
determines (3, when k is given. Both the formulas involve transcendent quan- 
tities ; they are nevertheless of great practical utility in this theory ; and they 
3 b 2 
