ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 283 



theorem maybe applied.' The representation of these functions, however, 

 •was not forthcoming, and later the solution of this problem was set as a 

 prize question by the Berlin Academy of Sciences. 



Abel had shown that the elliptic function a;=sn ^^, could be represented 

 as the quotient of infinite products. Jacobi, with the thought of repre- 

 senting an infinite product by means of a transcendental function, intro- 

 duced into analysis the so-called ti-function, which represents such a 

 product in the foi-m of a power series. Investigating further this tran- 

 scendent, he discovered its marvellous properties, and made use of it in his 

 further researches in the elliptic functions. Jacobi ^ founded the whole 

 theory of elliptic functions upon this new transcendent, which made these 

 functions remarkably clear and simple, as well as their applications, for 

 example, to rotatory motion, the swing of the pendulum, and innumerable 

 problems of physics and mechanics ; also by it the realms of geometry 

 were essentially widened, and many abstract properties of the theory of 

 numbers were revealed in a new light. 



Hence it appears that the ^-function showed itself of paramount 

 importance for the study of mathematics during the Jacobian epoch, and 

 as a prototype it served for the future development of the function-theory 

 and of all mathematics. 



(44) The elliptic function x=sn.u, as shown by Jacobi, maybe expressed 

 as the quotient of two 6-f unctions, where the ^-function may be written 

 in the form 



6 (u)^ :s 





in which m takes all integral values fi'om — co to -f cx3, m is the variable, 

 and the constant t is determined from the two pei'iods of the integral 



u 





where X is of the fourth degree in x ; or, as Jacobi says, t determines the 

 modulus of the elliptic integral. 



(45) 6-functions of two arguments. — Goepel,^ and in an independent 

 manner Rosenhain,^ generalised the simple ^-function of one variable and 

 formed analogous transcendents, the ^-functions of two variables 



+ 111 

 G{u,v)= JZi ' 



— !xin,n 



where here both vi and vi take all possible integral values from — ooto -f- oo, 

 u and V are the variables, and the constants aj, a2, and a^ are deter- 

 mined from the four periods of the integrals 



r dx , F xdx 

 I — = and — ^, 



JoVX Jo ^/^ 



' Jacobi, Werke, bd. ii. p. 517. 



^ Jacobi, Fund. Aova, p. 45 ; also Werke, bd. i. , p. 497. More recently Schellbacb 

 has made the fl-f unction his starting-point in his book, Die Lehre von elliptischen 

 Integralen und den Theta-Fwnctionen, Berlin, 1864. 



* Goepel, Theoria transoendentiiim Ahelianarum primi ordinis adumhratio levis 

 (Crelle, bd. xxxv. p. 277, 1847). 



• Rosenhain, Mevtoire sur Ics fonctions de deux variables a quatre j^criodes, S;c., 

 Mem. des Savants etrangers, t. xi. p. 361 ; see also Crelle, bd. si. p. 319. Further 

 see Jacobi, Notit Uher A. Goepel, Crelle, bd. xxxv. p. 313. 



