42 THE UNIVERSITY SCIENCE BULLETIN. 



Let f{w, z) ^0 he an irreducible polynomial in the two 

 complex variables w and z, w^ith either real or imaginary con- 

 stant coefficients. Substituting w =^u -\- iv and z^ x -\- iy in 

 the above relation we obtain the equation, 



P (x,y,u,v) -{-iQ (x,y,u,v) =0 (1) 



Whence, 



P (x,y,u,v) =0 (2) 



Q (x,y,u,v) =0 (3) 



The last two equations represent real three dimensional mani- 

 folds in the real four space (x, y, u, v) . Their intersection in 

 four space will be the surface $. Assume that w ^w^ when 

 z = Zq. It is then possible, in the neighborhood z^^, w^, to ex- 

 pand {w — w^) in powers of {z — z^) and by analytical con- 

 tinuation to go from the neighborhood of z^ to the neighbor- 

 hood of z^. As z changes from z^, to z^, w will change from Wq 

 into one of the values w^ corresponding to z^. If this process 

 be continued until z by a continuous succession of values re- 

 turns to z„, w may or may not return to w,,. In the first case 

 the representative point on $ corresponding to a pair of values 

 {w, z) will describe a closed path, while in the second case the 

 path will be open. The obvious one to one correspondence 

 between points of the surface $ and sets of values {w, z) shows 

 that this surface can play the same role as the ordinary Rie- 

 mann surface. 



If between equations (2) and (3) v is eliminated there 

 arises the relation, 



F {x,y,u)=Q (4) 



which represents in the three space {x, y,u), a surface F, viz., 

 the projection of <J> in that space. This surface F, as well as 4>, 

 can be used as a Riemann image, this being the configuration 

 to be investigated in this paper. We shall limit ourselves, as 

 before stated, to the hyper-elliptic case. It is evident that the 

 X, y or u projection of $ would serve the same purpose as F. 



Before proceeding with the general cubic a special cubic will 

 be considered in detail, and enough of the resulting surface 

 constructed to show its properties as a Riemann image. (This 

 special cubic is chosen on account of its adaptability to cross- 

 section representation.) 



