THE KANSAS UNIVERSITY 

 SCIENCE BULLETIN. 



Vol. XIIL] MAY, 1920. [No. 4. 



A Special Riemann Surface.* 



BY H. H. CONWELL. 



(Plates II to V.) 



THE purpose of this paper is to consider in detail, for 

 elliptic functions and briefly for hyper-elliptic functions, 

 a special Riemann surface in three space obtained as the pro- 

 jection of the intersection of two hyper-surfaces in four space. 

 It will be seen that the surface investigated here is of ad- 

 vantage in the fact that it can be easily identified, from the 

 point of view of analysis situs, with a double-faced disk hav- 

 ing p holes; where p =[-f-^]t, ^ being the degi-ee of the 

 function. In Riemann's real representation this is obtained 

 only after an artificial and somewhat complicated dissection 

 of the surface, in which the determination of the branch points 

 is a very important factor. In a sense this difficulty may be 

 said in our case to have been merely shifted from such a dis- 

 section to the construction of a certain real surface from its 

 equation in three space. This construction can, however, be 

 made very simple. In the ordinary Riemann surface the 

 actual location of the branch points is difficult at best, and is 

 useless so far as the investigations bearing on the surface are 

 concerned. The actual construction of the surface under con- 

 sideration will be avoided except in the simplest case, and 

 then only as much of its outline as is necessary will be ob- 

 tained. This construction will be found to be comparatively 

 simple. 



* Received for publication on April 29, 1920. 

 trn-in fi - 1 



L — 2 — J *^ understood to mean the greatest integer in — 5 — . 



(41) 



