92 UNIVERSITY OF COLORADO STUDIES 



3. Steiner''s Theorem. On a cubic C3 take two points v, and 

 Vj. Through %\ draw a straight line cutting C, in w, and ti^ Through 

 u^ and Vj draw a line cutting C^ in -w,; again through u^ and v, draw 

 a line cutting C^ in 'u^\ and so forth. The problem is to locate v^ in 

 Buch a manner that the 272th side passes through u^. The conditions 

 evidently are: 



^i + % + ^6 = 0, Vj+Wj+t/, = 0, 



Adding 



or 



?;,='yj-| (2wmj4-2w1j?/7,). (25) 



Hence, if -y, is given, Vj is determined many-valued. If one of 

 these values of -y, is known from (25), all vertices of the polygon 

 may be determined as soon as one, say u^^ has been arbitrarily 

 assumed. This result may be stated in the theorem, due to Steiner: 



The vertices of a polygon of 2n sides may he m,oved on a cubic 

 in such a manner that its odd sides pass through one fixed point v^, 

 while all the even sides pass through a certain fixed point -y,. 



4. A Problem of Closure admitting of a limited number of 

 solutions. 



From any point u^ of a cubic draw a tangent to it and let u.^ be 

 the point of tangency; from u^ draw a new tangent with the point of 

 tangency u^^ and so forth. In this manner a series of points 



^„ t/j, w„ . . ., ^„+, 



is obtained. The problem is to find the condition that 7/^^, coincides 



