268 ON THE RECENT PROGRESS OF ANALYSIS. 



He next shows, in a very remarkable manner, that, cor- 

 responding to a transformation in which we pass from a modulus 

 Jc to a modulus X, there exists another, whose formulae are 

 derivable from those of the former, in which we pass from a 

 modulus Vl k z to a modulus Vl A/ 2 , or which connects 

 moduli complementary to \ and Jc. The latter is accordingly 

 called, with reference to the former, the complementary trans- 

 formation. The first real transformation of k corresponds to 

 the second real transformation Vl k\ and vice versa. 



The next theorem which M. Jacobi demonstrates is not less 

 remarkable. It is that the combination of the first and second 

 real transformations gives a formula for the multiplication of the 

 original integral, or, in other words, that the modulus of the 

 integral which results from this double transformation is the 

 same as that of the original integral, so that the two integrals 

 differ only in their amplitudes. Of this theorem he had in the 

 earlier part of the work proved some particular cases*. 



After fully developing this part of the subject, he next 

 treats of the nature of the modular equation, and shows that 

 it possesses several remarkable properties. One is, that all 

 modular equations, of whatever order, are particular integrals 

 of a differential equation of the third order, of which the 

 general integral .can be expressed by means of elliptic tran- 

 scendents. 



16. We now enter on the second great division of M. 

 Jacobi's researches, the evolution of elliptic functions. 



* It may be shown that if we pass from k to X by the first transformation, we 

 can pass from A/I - X 2 to A/I -& 2 also by the first transformation. Also, as has 

 been said, we derive from the transformation {k to X} a transformation {^/i -k* 

 to v/i-X 2 }, and similarly from {*/! -X 2 to A/I -& 2 } a transformation {X to k}. 

 The first and last of these transformations correspond respectively to the diffe- 



rential equations 



dy _ i dx 



Hence, combining these equations and integrating, 



and it may also be shown that TT-TT, is an integer. 



