APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSUKE 91 



The two cases must be distinguished where n is odd or even. If n is 

 odd and adding in (23) the congruences with odd indices, then sub- 

 tracting from this sum the sum of the congruences with even indices 

 we obtain: — 



n-l n-l 



~ 2 



I 



In this congruence it is evidently impossible to make '«*„+! ="^1, for 

 any value of te^. In other words the problem cannot be solved for 

 an odd n. 



In a similar manner we obtain for our even n: 



a n-2 



7 2 



1 



which for '^0+1=^1 becomes: — 



n n-2 



2 2 



2r,,-2^,,+, = 0, (24) 



1 



a congruence which cau always be satisfied by choosing the v^b prop- 

 erly. If n — 1 of the v's are arbitrarily chosen, the nth v is by (24) 

 uniquely determined. Hence the theorem: — 



The vertices of a polygon of 2n sides^ A^ A^ . . . A^^, may he 

 moved on a plane cubic in such a manner that all its sides A^A^^ 

 A^ A„ . . . , ^2n -'^1 ^^*'^ about fixed points of this curvt. One of 

 these fixed points is uniquely determined by all the others. 



From (23) it is easily found that: — 



etc. 

 Hence the corollary: 



Also the sides A^A^^ A^A^^^ • • • 5 -^i-^zn/ A^A^^ A^A^, . . . / ^j„.s 

 A^^ turn about fixed points of the curved 



(>) See Kottbe: Die Entwickelung der synthetischen Geometrie. Vol. I. p. 153. 



