lie GEOMETRICAL RESEARCHES, 



positions for first extremities of inscribable closed 2 w'gons whose 

 first n sides and whose second n sides pass in order through the 

 n points. 



Hence (see theorem 9) we infer the following : 



THEOREM 23. 



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

 (n being an odd number) in one straight line, then will the problem 

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

 through the n points be partially porismatic ; and any point in tlie 

 surface will be an answerable position for the first extremity of an 

 inscribable closed 2 rig on whose first n sides and whose second n 

 sides pass in order through the n fixed points. Hip 33 It is evident 

 that when w=l, the locus of the extremities of the inscribable 

 1'gons will be the trace of the polar plane of the point through 

 which the sides all pass. 



15. The following theorem (of which Pascal's is but a parti- 

 cular case) is an evident consequence. 



THEOREM 24. 



If n be an odd number and that in a surface of the second degree 

 there be inscribed a closed 2 iCgon such that all its pairs of opposite 

 sides, with the exception of one pair, cut each other in n 1 points 

 lying in one straight line, then will this remaining pair of opposite 

 sides cut each other in a point in the same straight line. 



16. If n 2 be an even number, and that the inscription of the 

 closed (n 2)'gons whose sides pass in order through n 2 fixed 

 points in one straight line xx is non-porismatic ; then (see 

 theorem 14) by assming any position in xx as a n 1 th point, we 

 can find a corresponding position for a w tb point in the same line 

 so as to render fully porismatic the problem of the inscription of 

 the closed w'gons whose sides pass in order through the n points 

 of the series . 



