ON THE RECENT PROGRESS OF ANALYSIS. 269 



The evolution of elliptic functions into continued products 

 with an infinite number of factors presents itself as the limit 

 towards which M. Jacobi's theorem for the transformation of the 

 nth order tends as n increases sine limite. It is for this reason 

 that we may look on the problem of transformation as the lead- 

 ing idea in M. Jacobi's researches. 



We may in some degree illustrate these evolutions by a 

 reference to circular functions. A sine- is, as we know, an 

 elliptic function whose modulus is zero. Now if k is zero, X is 

 also zero. Thus if we apply a formula of transformation to a 

 sine, we shall be led to another sine either of the same or of 

 a multiple arc. Accordingly the first real transformation de- 

 generates in the case in question into the known formula for 

 the sine of a multiple arc ; while the second, leading us merely 

 to the sine of the same arc, becomes illusory. Thus in the case 

 of a sine, transformation is merely multiplication; but from 

 the formula for multiplication, viz. 





we at once deduce, by making (2m + 1) 6 = cf> and 2m + 1 in- 

 finite, the common formula 



This then is a formula of evolution deduced from the first 

 real transformation. It is however only when k is zero that the 

 first transformation will give such a formula. In all other cases 

 it is, for a reason which we cannot here enter on, impossible 

 to derive from it a formula of this kind. M. Jacobi's formulas 

 are accordingly derived from the second real transformation, and 

 therefore are illusory when k is zero, or for the case of the 

 sine. There is nothing therefore strictly analogous to them in 

 the theory of angular sections. By means of them we express 

 the function sin am x in terms of sin mx, m being a certain 

 constant. 



From the fundamental expressions in continued products, of 

 which there are three, many important theorems may be derived. 

 This part of the subject seems to admit of almost infinite 

 increase, and it is difficult to give any general view of it. 



