APPLICATIONS OF ELLIPTIC FUNCTIONS TO PKOBLEMS OF CL08UBE 105 



Thus, if ^1 is given, 



_ Imh -f- 2'm,^j — nu^ 



(-2d) 



n 



) • (2*) 



(2m^ + 2i7iiJ{,^ — nu^ 

 

 n 



This equation admits of n^ solution if n is prime, and includes x^. 



If c and u^ are determined according to (22) and (23), the 

 polygon on the corresponding hyperboloid always closes. We have, 

 therefore, the theorem : — 



If 11 is prime, then there are n? — 1 hyperboloids passsing through 

 the curve F, for each of which the polygons of 2n sides formed iy 

 its generatrices and inscribed to the curve are always closed. 



3. Steiner's theorem may easily be derived from this theorem. 

 Taking a point P on the curve F as a center of projection, and through 

 P the two generatrices PA and PB on one of the hyperboloids H, 

 determined by (22), and projecting P and all of the closed polygons 

 upon H then the points A and B are projected into two fixed points 

 V, and ^2 of the projected F, which is a plane cubic Cj. In the plane 

 a closed polygon of 2n sides is obtained whose sides alternately pass 

 through Vi and v^. 



Similarly Steiner's generalized theorem results from the first 

 theorem of this section by making a central projection with P as a 

 center'. 



§9. Pkojeotive Theory of Problems of Closure on the 

 Cubic. 



1. Comparing geometrical and analytical methods by which 

 problems of closure may be solved it is apparent that, analytically, 

 theorems of closure may be proved with great simplicity and elegance. 

 They all appear as special applications of Abel's great theorem, or 

 more specifically as applications of elliptic functions. 



The difficulty, however, arises when an attempt is made to 

 actually exhibit the results obtained by this method. It can be done 



(1) KOTTEE, lOC. Cit. 



