1V2 GEOMETRICAL RESEARCHES, 



THEOREM 13. 



If there be a surface of the second degree and n points such as to 

 render non-porismatic the problem of the inscription of the closed 

 rigons whose sides pass in order through the points ; then by assum- 

 ing any point in the straight line containing the first extremities of 

 the two inscribable closed rfgons as the n + 1 th point of the series, 

 we tvill render partially porismatic the problem of the inscription of 

 the closed (n + 1) 'gons whose sides pass in order through the n + 1 

 points. 



THEOREM 14 (porismj. 



If there be a surface of the second degree and n. fixed points, such as 

 to render non-porismatic the problem of the inscription of the closed 

 rUgons whose sides pass in order through the points ; then, assum- 

 ing any point in the straight line xx containing the first extremities 

 of the two inscribable closed n'gons, as the 11 + 1 th point of the series, 

 we can find a position for the n + 2 th point of the series in the same 

 straight line which will render fully porismatic the problem of the 

 inscription of the closed (n + 2) 'gons whose sides pass in order 

 through the n + 2 points of the series. 



8. And from theorem 14 we at once infer the following 

 theorems : 



THEOREM 15. 



If an open n'gon be inscribed in a surface of the second degree so 

 that its n sides pass in order through n fixed points, and that the 

 problem of the inscription of the closed rfgons whose sides pass in 

 like manner through the points is non-porismatic ; then will the 

 closing chord of the open rigon be in plane with the straight line 

 xx which contains the first extremities of the tiuo inscribable closed 

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



THEOREM 16. 



If in a surface of the second degree there be inscribed an open 

 2 rfgon whose two successive series of n sides pass in order through 

 n fixed povnts ; then will the plane containing its extremities and 

 the first point of its n 1 th side pass through the two answerable 

 positions for the first extremities of the inscribable closed n'gons 

 whose sides pass in order through the n points of the series. 



