HO GEOMETRICAL RESEARCHES, 



this we learn that the closing chords of all the inscribable open 

 rz-'gons must pass through the straight line zz of intersection of 

 the planes Aj and B r 



Now, if we conceive a plane through zz and the line G I c. l9 it 

 follows that all the inscribable open w'gons (whose sides pass 

 in order through the points) whose first extremities are in the 

 trace of this plane, will have their final extremities in the same 

 trace. And as the extremities of the %'gons form homographic 

 divisions in the trace, and that two distinct corresponding points 

 Cj and c n are interchangeable in these divisions, therefore it 

 follows that all the closing chords of these inscribable open w'gons 

 will pass through one point v (in the line zz). Moreover, it is 

 evident that the points of contact of the tangents from v to the 

 trace are answerable positions for first extremities of inscribable 

 closed %'gons whose sides pass in order through the n fixed points. 

 Hence, from this and theorem 6, we infer 



THEOREM 9. 



If there be a surface of ike second degree and n fixed points, such 

 that one closed 2 rigon can lie inscribed whose two successive series 

 of n sides pass in order through the points and are not coincident ; 

 then will any point in the surface be an answerable position for the 

 first extremity of an inscribable closed 2 n'gon whose sides pass in 

 like manner through the points ; and the problem of the inscription 

 of the closed nfgons whose sides pass in ordef through the n fixed 

 points is partially porismatic. 



THEOREM 10. 



If there be a surface of the second degree and a series of n fixed 

 points such that the problem of the inscription of the closed logons 

 whose sides pass in order through the n points is fully porismatic ; 

 then any one of the points of the series being omitted will render 

 partially porismatic the problem of the inscription of the closed 

 (n \.y gons whose sides pass in the same order through the n 1 

 remaining points. And according as the omitted point is inside or 

 outside the surface so will the closed (n lygons be imaginary or 

 real. H^f The trace of the polar plane of the omitted point 

 being the locus of an extremity of a side of the (n l)'goris. 



