46 THE UNIVERSITY SCIENCE BULLETIN. 



oval passes^through the two smaller roots of ( 15) . Let r\,r2,r-s, 

 be the roots of ( 15) , where rs > r2 > ri ; then n + r2 + rs = and 



— n r-2 n = q. From the last relation and the fact that ^ V ^^ 

 > g it is evident that o v ^^ > 2 r^i and therefore 'V^ > r2 ; in 



other words, x = V? does not lie within the oval. 

 3 



For x> ^-^ there are other maximum and minimum points or 



double points than for y equal zero. As in the simpler case these 

 sections are parabolic in nature. 



These investigations show that this surface has no impor- 

 tant characteristics, from our point of view, not common to 

 the rr >re special case and is therefore always reducible to a 

 double^^faced disk with one hole. 



THE GENERAL ELLIPTIC CASE. 



Up to this point the investigations have been confined to 

 the type, w- = z^ — pz -\- q, where p and q were both real, p 

 positi/e and the roots all real. It will now be shown that no 

 generality is lost by this restriction. 



Consider the general elliptic case, 



w- = / (z) (20)> 



where 



f{z) =a,(z — }\)(z — r,)(z — r,)iz — r,) ...(21) 



and a„ ?\, r^, i\, r^, are real or imaginary constants. The 

 elliptic integral resulting from this form may by a well known 

 transformation of fiz) be made to depend upon an integral of 

 the type, 



g(z) =b,(z-^ — a,z — a,).^- (22) 



No generality is therefore lost by replacing f{z) by g(z). The 

 constants of (22) may be positive or negative, real or imag- 

 inary. If a, and a^ are arbitrarily changed the surface F will 

 undergo a deformation. The only matter of interest in the 

 present paper is whether such a deformation increases or de- 

 creases the number of holes in F. It is of course evident that 

 if the number of holes is diminished as a., and a^ assume the 



* Beehm, Elliptische Functionen, Zweiter Teil, page 128. 



