BY MARTIN GARDINER, C.E. 107 



THEOREM 2. 



If in a surface of the second degree n'gons be inscribed whose 

 first extremities are all in one plane, and whose sides pass in order 

 through n fixed points, then will their last extremities be all in one 

 plane. 



THEOREM 3. 



If in a surface of the second degree there can be inscribed 3 

 closed logons whose sides pass in order through n faced points, then 

 will any point in the trace of the plane containing their first extremi- 

 ties be an answerable position for the first extremity of another such 

 inscribable closed n'gon. 



3. If in addition to having 3 inscribed closed ?i'gons, whose 

 sides pass in order through the n fixed points, we were to have 

 another such inscribed closed n'gon whose first extremity 1 is 

 not in the trace of the plane containing the first extremities of 

 the other three, then obviously any point y l in the surface is an 

 answerable position for the first extremity of a closed n'gon 

 whose sides pass in order through the n points. For through 

 x and y l we can conceive a plane whose trace cuts the trace of 

 the plane containing the first extremities of the other three closed 

 w'gons ; and therefore, from theorem 3, it follows that y 1 is an 

 answerable position for the first extremity of an inscribable 

 closed Ti'gon. Hence we have 



THEOREM 4. 



If in a surface of the second degree there can be inscribed 4 

 closed n'gons whose sides pass in order through n fix.ed points, and 

 that the first extremities of these closed rCgons are not all in one 

 plane, then will any point in the surface be an answerable position 

 for the first extremity of another such inscribable closed n'gon. 



4. Again (as a consequence from theorem 1) we have the 

 following : 



