292 ON THE RECENT PROGRESS OF ANALYSIS. 



we find some interesting remarks on the forms assumed by the 

 general transcendent when the biquadratic polynomial in the 

 denominator has four real roots. Dr Gudermann points out the 

 existence of a species of correlation between pairs of values of 

 the variable. 



23. The development of the elliptic function < in the form 

 of a continued product may be applied to establish formulae of 

 transformation. This mode of investigating such formulae was 

 made use of by Abel in his second paper in Schumacher's 

 Journal, No. 148, which we have already noticed; and a cor- 

 responding method is mentioned by M. Jacobi in one of the 

 cursory notices of his researches which he inserted in the early 

 volumes of Crelle's Journal. Mr Cay ley, in the Philosophical 

 Magazine for 1843, has pursued a similar course. Another 

 and very remarkable application of the same kind of develop- 

 ment consists in taking it as the definition of the function <, 

 and deducing from hence its other properties. It has been re- 

 marked that the continued products of Abel and M. Jacobi 

 are derived from considerations which, although cognate, are 

 yet distinct; those of the latter being singly infinite, while 

 Abel's fundamental developments consist of the product of an 

 infinite number of factors, each of which in its turn consists of 

 an infinite number of simple factors. Thus we can have two 

 very dissimilar definitions of the function < by means of con- 

 tinued products. M. Cauchy, who has investigated the theory 

 of what he has termed reciprocal factorials, that is, of continued 

 products of the form 



{(1 +x) (1+te) } {(14- faf) (1 + fx*) } 



which is immediately connected with M. Jacobi's developments, 

 has accordingly set out from the singly infinite system of pro- 

 ducts, and has deduced from hence the fundamental properties of 

 elliptic functions (Comptes Rendus, XVII. p. 825). 



Mr Cayley, on the other hand, has made use of Abel's doubly 

 infinite products, and has shown that the functions defined by 

 means of them satisfy the fundamental formulae mentioned in 

 the note at page 266, which, as these equations furnish a sufficient 

 definition of the elliptic functions, is equivalent to showing that 

 the continued products are in reality elliptic functions. He has 



