42 THE UNIVERSITY SCIENCE BULLETIN. 



Let f{w, 2:) = O be an irreducible polynomial in the two 

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

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

 the above relation we obtain the equation, 



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



Whence, 



P ix,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 = Wq when 

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

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

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

 hood of z^. As 2; 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 2; by a continuous succession of values re- 

 turns to ^o, IV may or may not return to w^. In the first case 

 the representative point on <l> 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) =0 (4) 



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

 the projection of <I> in that space. This surface F, as well as #, 

 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.) 



