m 



46 THE UNIVERSITY SCIENCE BULLETIN. 



ovalfpasses|[through the two smaller roots of (15). Let ri,r2,n, 

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



— nnrs = q. From the last relation and the fact that ^ P s 

 > g it is evident that g p ^^ > 2 rh and therefore \| > r2 ; 

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



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 more 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 

 positive and the roots all real. It will now be shown that no 

 generality is lost by this restriction. 



Consider the general elliptic case, 



w- = f (z) (20) 



where 



f(z) =a,{z~ J\) (z — r,) (z — r,) (z — 7',) ... (21) 



and a„» ^'i. *"2> ^'s^ ^'4» ^^e real or imaginary constants. The 

 elliptic integral resulting from this form may by a well known 

 transformation of /(z) be made to depend upon an integral of 

 the type, 



g(z) =bAz' — a,z — a,)^ (22) 



No generality is therefore lost by replacing / (2;) by g (z) . The 

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

 inary. If tto and ttg are arbitrarily changed the surface F will 

 undergo a deformation. The only matter of interest in the 

 present paper is vv^hether 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 



* Boehm, Elliptische Functionen, Zweitci- Teil, page 128. 



