MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 371 
the expression of a circular arc is made to pass into a logarithm ; so that the 
whole of this analytical theory rests on one principle, namely, the analogy 
which an elliptic function bears to a circular arc and to a logarithm, which 
are its extreme limits. 
10. Of the transformations in the last §, the principal one is contained in 
the formulas (1/), which constitute what is called the second theorem of M. 
Jacobi. One of its chief uses is to supply the defect of the first theorem by 
furnishing a direct process for reducing an elliptic function to a logarithm, 
through a scale of increasing moduli. In the function F ( h , <r), the modulus 
h being given, we know h! (= n/1 — h 2 ) named the complement of h for the 
sake of abridging ; we shall therefore obtain the amplitudes p 2 , &c., by the 
subdivision of the function H' = F ; we next compute the quantities 
j3' and H by the formulas (16) ; and, the amplitude <r being given, we deduce 
from the formulas (17), the amplitude r, which will satisfy the equation, 
F(A,»)=^.F(A,r), 
the modulus k of the new function being greater than h , because the comple 
ment k 1 is less than the complement h'. Taking now k' the complement of k, 
we deduce from it, by means of the formulas (16) and (17), the three quanti- 
ties (3/, kj, t p in like manner as (3', k', r were deduced from h ! ; and we shall 
have these equations, 
F (A, r) = (3/ F (A„ r,) ; F (h, ») = i X F (A, r,) ; 
the modulus k t being greater than k, because the complement kj is less than 
k'. The like operations being continued, we shall at length arrive at a mo- 
dulus k n , as near the limit 1 as may be required. 
Another use of the second theorem, when combined with the first, is to find 
any multiple of an elliptic function, or any aliquot part of it. By the first 
theorem, we have 
F (h, ^) = (3 F (k, <p) ; 
and by the second, making a = %|/ in the equations (17), 
F(*,r) = p'F(A,^) ; 
