MR. IVORY' ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 377 
and hence, 
1 — k 2 sin 2 X 2 — c 2 2 == j2 - 
And the formula (10) will determine when <p is given. 
Finally, therefore, we have these determinations, 
„ . ¥ (ty — <p\ 
(3 = 1 + 2 s, h = ^ + g y , tan ) = e tan <p ; 
F (*,?) = 
the modulus A being less than A 3 . By repeating the like operations, a suc- 
cession of moduli rapidly decreasing may be formed, by means of which the 
given elliptic function will be reduced to a circular arc as near as may be 
required. 
3 c 2 
