conwell: a special reimann surface. 45 



such passages and they be in opposite directions. Hence any 

 closed circuit on F can be reduced to zero or to sums of mul- 

 tiples of two irreducible circuits. These facts show the elliptic 

 function to be doubly periodic over F. 



THE general elliptic CASE FOR WHICH f {z) HAS REAL ROOTS. 



We shall now extend the preceding discussion to a general 

 elliptic function of the type 



IV- = z^ — pz -\r q (14) 



where p is positive and q either positive or negative, and where 

 the roots of 



z^ — VZ + q = (16) 



are all real. It will be shown that the resulting surface 

 F {x, y,u) =0 has properties identical with those of the 

 special case already investigated, if judged from the point of 

 view of the investigations of this paper. 

 We obtain at once, as in the preceding case, 



F {x, y, u) = Au^ — 4m-S — T- = (16) 



where 



S = x^ — 2>xy- — vx + q (17) 



and 



T = Sx^y — y- — py (18) 



The similarity of the XU and XY traces to those in the pre- 

 ceding case is obvious. From (16) we obtain, 



du ]/2y[-6x(S^-r)y^-\-Sx'-\- 6x''y' -6qx -\-'Sy' + 4p y'' - p' 



'^y~ A. (S-- T^)y^ [S + (5^ + T^)y^ y/^ . 



it 11 



For ^ = 0, — = for all values of x except the roots of x^— px + 

 q = 0, where it is infinite. For all negative values of x =^~ , 



J- is positive or negative for values oiy > 0, according to whether 



u is positive or negative, and negative or positive for ?/ < ac- 

 cording as u is positive or negative. Hence for all sections parallel 



to the y U plane, where x = \-J^ there will be a maximum and 



minimum point for y equal zero and for no other finite value of y . 

 Since the sum of the roots of (15) are zero, at least one root must 

 be negative and at l6ast one positive. It is also evident that the 



