conwell: a special reimann surface. 43 



Consider the equation 



w- =: z'^ — Slz — SO (5) 



from which 



p = u- — v^—(x^ — Sxy^ — 31x — S0)=0 .. (6) 

 and 



Q = 2uv — (Sx-y — y^ — 31?/) =0 (7) 



The intersection of P ^ and Q = in four space is the sur- 

 face $. The V projection of $ in three space has for its equa- 

 tion 



F (x, y, u) = 4w^ — 4w- {x^ — 3x?/- — 



31a- — 30) — (3.r-2/ — ?/3 — 31i/)- = 0.... (8) 



This surface is symmetric to both the XU and XY planes. 

 The trace on the XU plane is the XX axis and the real curve 



u'^x^ — Slx — SO (9) 



representing all the real pairs {w, z) satisfying the original 

 equation. The curve represented by (9) consists of an in- 

 finite branch and an oval (see fig. I). The XY trace consists 

 of the XX axis and the hyperbola (see fig. II). 



Sx- — y^ = Sl (10) 



This hyperbola and the XX axis are the only double curves of 

 the surface. 



From equation (4) we obtain, 



u=^V^[s + {S'- + ry^ (11) 



where 



S = x^ — Sxy- _ 31a' — 30 (12) 



and 



T = Sxhj — y^ — Sly (13) 



In this expression for u only positive values of the inner radical 



are considered as only real points on the surface F are to be in- 



ou 

 vestigated. Investigations of ( 11) show that when y = 0,j^ =0 



for all values of x except 6, — 1 and — 5, where it is infinite. 



Vol O 77 



— „ _ ^and y > 0, ^r- is positive or negative ac- 

 cording to whether u is positive or negative, while for negative 

 values of y it is positive or negative according to whether u is 

 negative or positive. Hence for all sections of the surface parallel 



