MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 351 
happily conceived the idea of expressing the amplitude of an elliptic function 
in terms of the function itself. By this procedure the sines and cosines of the 
amplitudes become periodical quantities like the sines and cosines of circular 
arcs ; and analogy immediately points out many new and useful properties 
which it would be difficult to deduce by any other mode of investigation. This 
new way of considering the subject struck out by M. Abel, not only disclosed 
to him some interesting and original views, but it conducted him to the general 
and recondite theorems which, without his knowledge, had been previously 
discovered by the geometer of Konigsberg. M. Jacobi, following in his 
researches a different method from M. Abel, proved that an elliptic function 
may be transformed innumerable ways into another similar function to which 
it bears constantly the same proportion. In the solution of this problem the 
modulus and the amplitude sought are deduced from the like given quantities, 
by equations which depend upon the division into an odd number of equal 
parts of the definite integral, having its amplitude equal to 90°; and, as any 
odd number may be chosen at pleasure, the number of transformations is 
unlimited. In consequence of this discovery, an elliptic function can have its 
modulus augmented or diminished according to an infinite number of different 
scales. The new process for effecting the same reduction discovered by 
Legendre in 1825, is only the most simple case of the extensive theorem of 
M. Jacobi ; and, although the older transformation of Lagrange is no part of 
the same theorem, it bears to it a close resemblance in every respect. Such is 
the principal addition made to this branch of analysis by M. Jacobi ; but the 
new methods of investigation introduced by him and M. Abel, open a wide 
field of collateral research, which probably will long continue to furnish matter 
for exercising the ingenuity of mathematicians. 
But it seldom happens that an inventor arrives by the shortest road at the 
results which he has created, or explains them in the simplest manner. The 
demonstrations of M. Jacobi require long and complicated calculations ; and 
it can hardly be said that the train of deduction leads naturally to the truths 
which are proved, or presents all the conclusions which the theory embraces 
in a connected point of view. The theorem does not comprehend the trans- 
formation of Lagrange, which must be separately demonstrated. This is an 
imperfection of no great moment ; but it is always satisfactory to contemplate 
2 z 
MDCCCXXXI. 
