MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 373 
1 1 . We have now demonstrated, as was proposed, the principal and leading- 
points of this theory, for which we are indebted to M. Jacobi. For the sub- 
ordinate details, and for many curious and important collateral researches that 
have been suggested by the new views laid open in this branch of analysis, we 
must refer to M. Jacobi’s own work, to the papers of M. Abel, and to the 
writings of Legendre. We shall conclude this paper by applying the formulas 
that have been investigated to two particular instances, taking for p the most 
simple values, namely 2 and 3. 
Example 1. Supposing p = 2. 
By the formulas (3) and (6) we have these equations between the amplitudes 
4> and <p, z being = sin <p, 
sin %// = 
Vl — Z 2 
Vl 
l - 
C°s^= 
sirr A 
wherefore 
and 
We now get 
1 = _ *2) + (l ..-jJl-) 2 . 
sin 4 Ai ’ sur Aj 
- ( 3 2 = k 2 . 
= 1 - k 2 ; 0 = 1 +V; 
Also, by the formula (9), 
7 79*4-v k 2 l — y 
h — k- sin ^i — (j + 7 ; 
from which we deduce 
(1 + h)(l + V) = 2. 
The equation (1), viz. 
F(h,^)=pF(k,<p), 
may now be put in one or other of these two forms, 
F(k,p) = ±±*F(h,4), 
F (/>,+) = (1 + U) F (A, <p ) : 
