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UNIVERSITY OF COLORADO STUDIES 



a'' c 



(12) 



These represent two elliptical cylinders intersecting each other in the 

 given quartic, 

 A plane 



Aa;+By+Cs+D=0 (13) 



intersects this curve in four points («;, yj, s^ i=\^ 2, 3, 4. The 

 application of Abel's theorem, (10), to (12) and (13) gives 



dx, dx^ dx^ dx. 



— 1-1— l+—i+ —1:^0, 



yi^i y-^i Vz^z 2/«24 



or, after replacing «, y, z by their values extracted from (11), 



du^-{-d^l^-\-du^-\-du^z=.{)^ 

 '?^iH-Wj+'W3+'W«=const., (14) 



where the -w's are the arguments in (11) corresponding to the points 

 of intersection of the plane with the quartic. To determine the con- 

 stant in (14), let (13) coincide with the ;y3-plane. Then x=a2,nu=0 

 and from the theory of elliptic functions it is known that the sum 

 of the arguments for which 8nw=0 is = 0(mod. per.), or 



As every quartic in space may he transformed into the curve 

 (12) we have the theorem: — 



If a quartic in space of the first kind is parametrically repre- 

 sented hy elliptic functions of the argument u, then the su7n, of the 

 arguments of its points of intersection with any plane is congruent 

 zero^ modulus periodicity. 



§8. Problems of Closure. 



1. On a quartic V assume any two points P and v^ and pass a 

 plane through P and v^ cutting F in two other points -w, and ^j, then 



