CONWELL.: A SPECIAL RIEMANN SURFACE. 49 



representing all the real pairs (iv, z) satisfying the original 

 equation. The latter consists of two ovals and an infinite 

 branch. The trace on the XY plane is the double curve repre- 

 sented by the equation T- = 0. This curve is composed of the 

 XX axis and four infinite branches which are hyperbolic in 

 form and coaxial (see fig. VIII). 



Sections parallel to the XU plane give rise to curves which 

 have double points on the branches I and III of the double 

 curve, as shown in the figure, and nowhere else. This is 

 shown by an investigation of the value of S in the neighbor- 

 hood of these branches. For the two branches to hang to- 

 gether or intersect each other, it is necessary that T be equal 

 to zero and S be negative or zero. Every pair of values (x, y) 

 on one of these infinite branches reduces T to zero, but none of 

 these pairs on branch II or IV will cause 5 to be negative or 

 zero. Therefore the two sheets of the surface F do not cut 

 through each other along either of these branches. The two 

 sheets hang together along the XX axis from — oo to — 3, — 2 

 to — 1, from + 1 to +5 and cut each other along the two 

 branches I and III of T ^ 0. To prove, as in the elliptic case, 

 that the two sheets never hang together for any finite value of 

 y except zero would be very complicated, and so another method 

 is employed. It is easily seen that any section parallel to the 

 YJJ plane will give rise to a curve which has a number, say d, 

 double points. But this cuvve is composed of two branches 

 which intersect in d points in the XY plane. If d is odd the 

 two branches are odd and hence each branch stretches off to 

 infinity in both directions. If d is even, each branch is even 

 and hence cuts the line at infinity in an even number of places 

 and is accordingly a closed curve. In the first case {d odd) the 

 XX axis must be composed of intersection points, while in the 

 latter it is not. This leads to the conclusion that all sections 

 which cut the curve u^f{x),y^^O give rise to even branches 

 and all others to odd. Hence the former are always reducible 

 to traces of the form, fig. V or fig. VI, while the latter are 

 always reducible to branches of the form fig. VII. From this 

 will follow, as in the elliptic case, that F is two-sheeted and 

 contains two holes. By a deformation similar to the one de- 

 scribed in the example of the elliptic case, it may be brought 

 into the form of a double-faced disk with two holes. Hence all 



4 — Sci. Bui. — 860 



