conwell: a special riemann surface. 51 



In the elliptic case the double curves consisted of the XX 

 axis and an hyperbola. That part of the XX axis included by 

 the real part of the curve u = f(x), 2/ = is isolated. Of the 

 hyperbola, that branch lying to the left of the YU plane is 

 isolated. 



In the hyper-elliptic example the double curve consists of 

 the XX axis and four infinite branches. What v^as said of the 

 XX axis for the elliptic case holds here also. Of the four 

 infinite branches two are isolated (see fig. VIII), and two are 

 curves of intersection of the two sheets of the surface. 



The same conditions will exist in the general hyper-elliptic 

 case, the XX axis always being a double curve with the same 

 law as to isolated points as in the simpler cases. The other 

 double curves will be partly isolated and partly curves of in- 

 tersection of the two sheets of the surface. The isolated curves 

 separate themselves from the other class in that they always 

 pass through one or more of the ovals, while the curves of in- 

 tersection of sheets never do. 



