44 THE UNIVERSITY SCIENCE BULLETIN. 



to the Y U plane, where x ^ yH there will be either both a 

 ^ ' — 3 



maximum and minimum point, or a double point, for y equal 



zero and for no other finite value of y. For x > \-|^ , there are 



other maximum and minimum points and double points, and the 

 curves all pierce the X Y plane along the curve represented by- 

 equation ( 10) . From the preceding discussion and an inspection 

 of equation (9) and figure I it is evident that the orthogonal 

 projection of F upon the X U plane will be nowhere within the 

 oval, and hence that there is a hole in F for which the oval is the 

 central section. 



It is obvious that the surface F is composed of two sheets 

 (see figs. I-VII) which hang together along the XX axis from 

 — 00 to — 5, from — 1 to + 6 and pass through each other 

 along the branch of the double curve T =^=0 which lies to the 

 right of the YY axis. 



Sections parallel to the XU plane give curves composed of 

 two branches which cut each other in points on one branch of 

 the double curve T = and nowhere else. Each branch con- 

 tinues to infinity and there unites parabolically with the other. 

 The YU sections also unite parabolically at infinity, and hence 

 the two sheets of the surface F merge into each other every- 

 where at infinity. 



The surface F may be reduced to a double-faced disk with 



one hole as follows: For all values of a: > \— - deform the 



surface by pulling the sheets through each other in such a 

 way that instead of cutting in two distinct points on T = 

 for each value of x they will cut each other in two coincident 

 points. This deformation will be continuous and approach 



zero in magnitude as x approaches x^- ^^^ will nowhere 



o 



produce a tear in the surface. Having made this deformation, 

 project the surface upon the XU plane and the result will be 

 a double-faced disk with one hole. 



Starting at a point P in sheet I and continuing in any direc- 

 tion on the surface F we can always return to P. This closed 

 path may be all in sheet / or in both sheets / and //. It may or 

 may not pass through or around the oval. In the latter case 

 the circuit can always be reduced to zero while in the former 

 it cannot be so reduced, unless there be an even number of 



