368 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
the modulus h t being less than the modulus h. Continuing the like operations, 
we can pass along a scale of decreasing moduli, till we arrive at one which, 
being as small as we please, will make the function F (Jc, q>) approach to a cir- 
cular arc as near as may be required. 
If we wish to apply the same theorem to reduce the given function F ( h , cp) 
to a logarithm, through a scale of increasing moduli, the process is not so 
direct. For, in the first place, the greater modulus k is not immediately de- 
ducible from the less h, by means of the formulas that have been investigated; 
and, in the second place, the amplitude <p cannot be found when is given 
without solving an equation of p dimensions. The theorem is, no doubt, 
mathematically sufficient for effecting the reduction; but the operations re- 
quired are practically impossible, except in a few cases when p is a small 
number. But the ingenuity of M. Jacobi has provided a remedy for this in- 
convenience by a new transformation, which we shall now briefly explain, as 
it discloses a new set of remarkable properties of the elliptic functions. 
If we put y = tan ■>£,# = tan <p, h' 2 = 1 — h 2 , Jc 2 = 1 — Jc 2 , the differential 
of the equation (1) will assume this form, 
dy /3 dx 
■v/ 1 + y 2 . I + h' 2 y- 4 / 1 + x 3 . 1 + k hl x 2 ’ 
and, for solving this equation, we shall have by the formula (7), 
p being an odd number, 
X 1 X- X~ 
tarr A 2 * tan 2 A 4 * tan 2 \ p — 1 
x l X° X 3 
tan 2 A, * tan 2 A 3 * tan 2 Xp — 2 
But if this value of y solve the differential equation, it will still solve it, if we 
change + x 2 and + y 2 into — x 2 and — y 2 ; for it is obvious that, if the ex- 
pression of y make the two sides of the equation identical in one case, it will 
necessarily make them identical in the other case. Wherefore the equation 
dy (S dx 
4/ 1 — y 3 . I — h* y l 4/I — x 1 . 1 — k Vi x 3> 
1 - 
y = (3x X 
1 — 
will have for its solution. 
