138 
ON THE INVERSE ELLIPTIC FUNCTIONS. 
[24 
depends in a remarkable manner on the mode in which the superior limits of to, n 
are assigned. Imagine to, n to have any positive or negative integer values satisfying the 
equation 
(4). 
<f> (to 2 , ri) < T 
Consider, for greater distinctness, to, n as the coordinates of a point; the equation 
0 (to 2 , ri 2 ) = T belongs to a certain curve symmetrical with respect to the two axes. 
I suppose besides that this is a continuous curve without multiple points, and such 
that the minimum value of a radius vector through the origin continually increases as 
T increases, and becomes infinite with T. The curve may be analytically discontinuous, 
this is of no importance. The condition with respect to the limits is then that to 
and n must be integer values denoting the coordinates of a point within the above 
curve, the whole system of such integer values being successively taken for these 
quantities. 
Suppose, next, v! denotes the same function as u, except that the limiting condition is 
0' (to 2 , ri 2 ) < T' 
(5). 
The curve 0' (to 2 , ri) = T' is supposed to possess the same properties with the other 
limiting curve, and, for greater distinctness, to lie entirely outside of it ; but this last 
condition is nonessential. 
These conditions being satisfied, the ratio ri : u is very easily determined in the 
limiting case of T and T' infinite. In fact 
(6), 
or 
the limiting conditions being 
0 (to 2 , ri)> T . 
0 / (to 2 , ri) < T. 
Now 
(10), 
or, the alternate terms vanishing on account of the positive and negative values 
destroying each other, 
(11). 
In general 
SX\fr (to, ri) = ff\[r (to, ri) dmdn + P 
(12),