130 UNIVERSITY OF COLORADO STUDIES 



,'-r ^^ (6) 



Jo V{\-z^) (l-/fcV) 



by which the positive part of the 2-plane is conformally mapped 

 into the rectangle of the s' -plane, whose sides correspond to the 

 periods of the elliptic function 



s = Bnz'^ (7) 



In order that the sides of this rectangle be w^ and w^, the modulus 

 ^ of (6) must be chosen 



,^ r(i+g^Hi±2lii±£)^T , 



'^* L(l+?) (!+?') (1+?')...J 



where ^=e ' . 



By this determination the periods of 2 = Qnz' become 27r and 



- > the general period 



The torus may be considered as a continuous deformation of the 



doubly -sheeted Riemann surface with the branch points ±1, ± — > and 



represents the surface for all doubly periodic functions with the 

 periods (8). 



If now in the z' plane, Fig. 13, two lines are drawn from the 

 origin O to two points of the period parallelograms; one to the point 



the other to the point 



(1) See F. Klein, "Uber Riemann's Theorie der AlGrebraischen Funktionen und ihrer Inte- 

 erale," pp. 50-55. 



