ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 251 



Another important discovery is due to Abel's investigations : when 

 the multiplier became infinitely large in the formulse through which he 

 represented the elliptic functions of a multiple argument by means of 

 functions of the simple argument, he obtained remarkable expressions for 

 the elliptic functions in form of infinite series that are expressed as 

 quotients of infinite products.' 



Jacobi, contemporaneously with Abel, was occupied in another part of 

 the theory of elliptic functions, and with equally as great success. A 

 fortunate induction of considering the transformation and the multiplica- 

 tion from a common point of view, and the last as a special case of the 

 first, led him to the conjecture that rational functions of any degree may 

 be used to transform an elliptic integral into an integral of the same form. 

 This conjecture was at once confirmed, since the number of constants, 

 which may be arbitrarily disposed of for any degree is sufiicient to satisfy 

 all conditions in order that the form of the transformed integral may 

 agree with the original {cf. Dirichlet, he. cit. p. 12). 



Jacobi also showed how the elliptic functions may be expressed in the 

 form of infinite products, which may be represented by trigonometric 

 series, and he further used the infinite series to express the square and 

 product of these functions. These results, with the general theory of 

 transformation, are systematised in the ' Fundamenta Nova,' Konigsberg, 

 1827; Jacobi's 'Werke,' bd. i. p. 49; further developments in this 

 direction are mentioned by Enneper ('EUip. Funct.' p. 74 et seq.). A 

 report of this work is given by Ellis (pp. 49-59). See also a paper by 

 Poisson (Crelle, bd. x. p. 342). 



(12). Statement of Abel's Theorem — We write, as above, 



If in this expression X = 1 — x-, and if there exists a given algebraic 

 relation x^^ -{■ x^^ ^ Ij then, 



(1) n(a;,) + n(a;2) = Constant ; 



i.e. if sin^^ + sin^i// = 1 then f i- 4' =k. 



Further, if X= (1 — a;-) (1 — K^a;^), and if we have given the algebraic 

 relation ^ 



4(1- x^^) (1 - Xi^) (1 - x^^) = (2 - cci* - Ka^ + x^^ + K^x^^x^WYf 

 then is 



(2) U(x,) + U{x,) + n{x,) = 0. 



From these two examples it is seen that, although in general we cannot 

 integrate (I.) by means of algebraic or logarithmic functions, neverthe- 

 less, we have expressions (1) and (2) for the sums of such integrals, pro- 

 vided the variables that occur in these integrals are connected by algebraic 

 relations. 



By Euler's theorem, any number of elliptic integrals of the first kind may 



' See in this connection Cayley, Liouv. Journ. x. p. 385 ; also numerous papers 

 in his collected works : — Heine (Crelle, bd. xxxiv. p. 122) ; Eisenstein (Crelle, bd. xxxv. 

 p. 153 ; Liouville {Lumv. Journ. t. ii. p. 433) ; Lipschitz {Acta Math. bd. iv. p. 193); 

 Biermann {Theorie der analytischen Functionen, p. 323), &c. 



* See a paper by Boole, ' On the comparison of transcendents,' Phil. Tram. 1857, 

 p. 750; and also Rowe, ' Memoir on Abel's theorem," Phil. Tram. Pt. III. 1881. 



