APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 93 



with u^. Designating multiples of periods by Ai, Aj, A3, . . . , A„ we 

 have : — 



u^-\-2us=A„ 



from which, if u^^i=Ui 



u^=A^—2u„ 



^n-i=A„.-2..„=A„.-2A„+2V„ 



^n-.=A„.,-2./„,=A„,-2A„.,+4A„-2V 



and 



u, = 2mw-\-2m{iMi^+ ( — 1 )°2°^^„ 

 2mw-{-2m^w^ 



u,=- 



(26) 



From this it is seen that in every case there are a limited num- 

 ber of solutions which depend upon the division -problem of elliptic 

 functions. For n=l the points of inflection are obtained. For 



mw2-\-2m'jWi 



u,=- 



9 



in which m and m^ may assume all possible values from to 8. 

 Among the 81 solutions are contained the 9 points of inflexion, so 

 that only 72 ordinary closed triangles of the prescribed kind exist on 

 the cubic. We have here an essentially different problem from those 

 with an infinite number of solutions. 



§4. Steiner's Couples and their Properties. 

 1. In §3 the equation 



