352 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
a theory in its full extent, and to deduce all the connected truths from the 
same principles. On a careful examination it will be found that the sines or 
cosines of the amplitudes used in the transformations are analogous to the sines 
or cosines of two circular arcs, one of which is a multiple of the other ; inso- 
much that the former quantities are changed into the latter when the modulus 
is supposed to vanish in the algebraic expressions. We may therefore transfer 
to the elliptic transcendent the same methods of investigation that succeed in 
the circle. When this procedure is followed, there is no need to distinguish 
between an odd and an even number ; the demonstrations are shortened ; and 
the difficulties are mostly removed by the close analogy between the two cases. 
It is in this point of view that the subject is treated in this paper, in which 
it is proposed to demonstrate the principal theorems without going into the 
detail of the applications. 
1 . Elliptic functions of the first kind are of this form *, viz. 
d <p 
V 1 — k 9 sin* <p 5 
V 1 — h 2 sin' 4> ’ 
the arcs <p and \p being the amplitudes, and the quantities k and h, which are 
always less than unit, the moduli of the functions. For the sake of abridging, 
I shall denote the foregoing integrals by K (<p) and H (\Js), the prefixes K and 
H having reference to the moduli h and h ; and, for the definite integral 
between the amplitudes 0 and^, I shall use indiscriminately either K (^j 
and H or, more simply, K and H. 
The general equation to be investigated is the following. 
dj> 
— h 2 sin 2 4/ 
dtp 
V 1 — k 2 sin 2 <p ’ 
( 1 ) 
/3 being a constant quantity equal to the first ratio of the nascent arcs and <p. 
* In what follows, the terms ‘ elliptic functions’ and * elliptic transcendents ’ are to be understood as 
applying to those of the first kind only, which alone are treated of. 
