A FEW EXAMPLES LN THE THEORY OF FUNCTIONS 249 



/(z') are permuted; i. e., whether the zero point of }(z') or the point 

 at infinity for the original function is a branch-point. In the Riemann 

 surface this is equivalent with describing a closed path including both 

 branch- points, in our example, and in all three sheets. Thus, starting 

 out with one determination of w, for instance w x , then after one com- 

 plete revolution of z in the surface, w z is permuted with w 3 , after a 

 second revolution w 3 is permuted with w 2 and after another revolution 

 w 2 returns with the original value w x . This path has been drawn in 

 the surface and appears in Fig. 5. 



From this it follows that z=oo is a branch-point of the function 

 w for all roots. 



(e) CONSTRUCTION OF MODELS FOR U AND V IN THE FUNCTION 



w=u+ iv 

 Eliminating in turn v and u from (3) and (4) we obtain 



8« 3 — 2U— x lu 3 —u+x /ns 



y=± <*.' (8) 



—8v 5 — 2V—y Iv^+v—y 



x=±— = — ^J — '- . (9) 



37; \ 37 



We can now consider u and v, in turn, each as constants and for every 

 value of u and v construct the curves (8) and (9) in two separate xy- 

 planes. If at the same time we give these curves distances vertically 

 above and below the rry-planes corresponding to the values of u and v, 

 two surfaces are obtained which for every value of z=x+iy give the 

 corresponding values of u and v of the function w as the perpendiculars 

 erected at 2 to the z-plane and bounded by the surfaces represented by 

 (8) and (9). 



These surfaces which are of the ninth order have not yet been modeled. 

 A number of models of this kind have been constructed by A. Wildbret, 

 H. Burkhardt, J. Kleiber, under the direction of Professor W. 

 Dyck, and are sold by Martin Schilling in Halle a/S, Germany. 



(f ) Expansion into Series 



When z turns twice around a branch-point, say B T , and the function 

 starts out with the determination w z , it will return with the same value. 



