ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 249 



and showed that it was always possible to reduce it to one or the other of 

 three forms essentially different. 



We may mention, in passing, as being among the early English con- 

 tributions to the subject memoirs by Brinkley ('Dublin Trans.' ix. p. 145, 

 1803) and Wallace ('Edinb. Trans.' v. p. 253). A criticism of Talbot's 



* Researches on the Integral Calculus '(' Phil. Trans.' 1836, p. 177, and 

 1838, p. 1) is given by Ellis, p. 41. 



(9) The theory of the elliptic functions, as Abel and Jacobi ^ found it 

 in 1827, offered many highly enigmatical phenomena, which could not be 

 explained by the principles that were at that time in vogue. For example, 

 the degree of the equation which is found by means of Euler's theorem, 

 and upon whose solution depends the division of the elliptic integral, 

 was not, as in the analogous question of the division of the circle, equal 

 to the number of the parts, but to the square of this number. It was easy 

 to see the meaning of the real roots, whose number agrees with the 

 number we have in the division of the circle ; however, the number of 

 imaginary roots must have seemed without explanation (Dirichlet, 



* Gedachtnissrede,' p. 9). 



We shall next consider the inverse functions of the integrals which we 

 have been treating. With Jacobi ^ we begin with the simple algebraic 

 integral 



P dx . _i 



M= =sm ^x. 



p dx 

 Jon/1-c 



In this expression we may either consider %<, as a function of the upper 

 limit X, or inversely, the upper limit a; as a function of u. In the first 

 case, when tt ^sin~^a;, it is not possible to express u in the form of a 

 power series which is convergent for every value of x ; and for a given 

 value of X, u is not determinate, but has an infinite number of values, 

 differing by multiples of 27r. But when we regard the upper limit a; as a 

 function of u, and write cc^sin u, then x may be expressed as a series 

 which is convergent for all values, real and imaginary, of u; and when u is 

 given a definite value, then x also has a definite value, and x considered 

 as a function of tt enjoys all the properties of a rational function. 

 The next more general algebraic integral is the elliptic integral 



M= r ^"^ =T\{x). 



As above, tt=n(a;) cannot be expressed by a series that is always 

 -convergent ; and for a given value of x the variable u has not a definite 

 value, but a double infinity of values, differing by multiples of the periods 

 of elliptic functions (see next article). 



The innate property of this integral could not be recognised if we 

 considered the transcendent x alone ; but we have to regard the upper 



' Their first writings on this subject are : Abel, Crelle, bd. ii. September 1827 ; 

 Jacobi, two letters to Schumacher dated June 13 and August 2, 1827, in the Attrono- 

 viiiche NachHchten, No. 123, vol. vi. 



- Jaoobi, Contiderationei generalei de tramcendentibus Abelianif ( Werke, bd. ii. 

 p, 8). 



