94 UNIVERSITY OF COLORADO STUDIES 



v^=V3-\ (2mw-\-2m^w^) (27) 



n 



has been established, by which two points v, and v^ on a cubic must 

 be related in order to serve as fixed points of an infinite number of 

 closed polygons. Any two points of this kind form a Steinerian 

 couple. 



To find the coordinates of v, when v^ is given, say v^= — , by 



(16) 



(a-\-2mw-\-2mM.\ ,„^, 



„ )• (2«) 



The value oi pi — j being given, x may be expressed in terms 



of this in 7^' different ways, as is well known from the division -prob- 

 lem of elliptic functions. Of these we have to exclude the case 

 where Vj^Vj, so that in reality only ri? — 1 pairs are left. It is clear 

 that this number is reduced when n is not an odd prime. The result 

 may be stated in the theorem: — 



To every point v^ of the curve are associated as many points i;„ 

 forming in each case a couple^ as there are pairs of numbers m, m,, 

 (<ri) which have no common factor with n. 



From formulas (20) and (27) the special theorem is evident: — 



Any two points of inflexion of a cubic may be the fixed points 

 of a Steinerian polygon of six sides. 



Taking the points of tangency of the tangents from any two 

 points of inflexion and applying formulas (22) it is found that these 

 points of tangency are the fixed points of a Steinerian polygon of 

 twelve sides. 



2. All couples for which m> and 7^, have constant values are 

 said to belong to the same class, which is characterized by the com- 

 bination m, Wj. To obtain all couples of the same class, when one 

 (v„ Vj) is known, connect any point v of the curve with -y, and v^ by 

 straight lines which cut the curve in two other points v/ and v/, re- 

 spectively. Then 



