BY MARTIN GARDIXER, C.E. 117 



This may be formally enunciated, thus : 

 THEOREM 25. 



Ifnbe any even number, and that we have a surface of the second 

 degree and n 1 points in one straight line ;' then a position for a n th 

 point can be found in the same straight line which will render fully 

 porismatic the problem of the inscription of the closed n'gons whose 

 sides pass in order through the n points of the series 



17. If there be n fixed points in the plane of a conic, we know- 

 that the problem of the inscription of closed Ti'gons in the conic 

 is either non-porismatic or fully porismatic ; and in the non- 

 porismatic state of the data we can always find a real line 

 containing the two answerable positions for the first extremities 

 of the closed n'gons. 



Hence we easily arrive at the following theorem. 



THEOREM 26. 



If a gauche closed rfgvn inscribed in a surface of the second 

 decree be cut by a plane, and that we conceive the points of its inter- 

 section with the plane to become fixed, then the problem of the 

 inscription of the closed n'gons whose sides pass in order through 

 these n fixed points is partially porismatic or fully porismatic, just 

 according as the problem of the inscription of the closed n'gons in 

 the trace of tJie plane containing the points is non-porismatic or 



18. Let the surface S and the n points o 1? o 2 , . . . . o n be so 

 related that the problem of the inscription of the closed w'gons 

 whose sides pass in order through the n points is non-porismatic. 



Let p l and q l be the first extremities of inscribable closed 

 w'gons ; and let xx be the straight line containing these points. 



Through xx conceive any plane cutting the surface S. Tow 

 if in the trace of this plane we assume points as first extremities 

 of inscribable open ^'gons whose sides pass through the n fixed 

 points, then will the final extremities of these w'gons be in the 

 same trace. And, as the extremities of each of these ^'gons 

 are corresponding points in homographic divisions in a conic 



