conwell: a special reimann surface. 43 



Consider the equation 



iv^ = z^ — Slz — SO (5) 



from which 



p = u^ — v^ _ (x^ _ sxy- — 31a; _ 30) = . . (6) 

 and 



Q = 2uv— (Sx-jj — r — 31?/) =0 (7) 



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

 face *. The V projection of <I> in three space has for its equa- 

 tion 



F (x, 7j, u) = 4u' — Au- {x^ — 3x?/2 — 



31a; — 30) — (3a;27/ — ?/ — 3l7y)2 = (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'' — Zlx — SO (9) 



representing all the real pairs {ii\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). 



3a:2 — 7/ = 31 (10) 



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

 the surface. 



From equation (4) v^^e obtain, 



u= ^V\[s + {S''-\-T~)y^''' (11) 



where 



S = xy" — Sxy- — 31a; — 30 (12) 



and 



T = Sxhj — qf — 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- 



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

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

 For values of x :^ V'^and y > 0, V- 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 



