BT MARTIN GARDINER, C.E. HI 



THEOREM 11. 



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

 i der partially porismatic the problem of the inscription of the 

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

 by adopting the pole of the plane whose trace is the locus of the first 

 extremities of the inscribable closed n'gons, as the n + 1 th point of 

 the series, we render fully porismatic the problem of the inscription 

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

 series of n + 1 points. 



THEOREM 12. 



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 non-porismatic ; 

 then there can be two and not more than two answerable positions 

 (real or imaginary as may be) for the first extremities of the cl> sed 

 ugons. 



7. Let us have a surface of the second degree and n points o^ o^ 

 .... o , such as to render non-porismatic the inscription of the 

 closed n'gons whose sides pass in order through these points. 

 And let x x be the straight line containing the first extremities of 

 these closed ?&'gons. 



If o be any point in the line x x, then evidently the 

 points in which xx pierces the surface are answerable positions for 

 first extremities of inscribable closed 2 (n + l)'gons each of which 

 has its two successive series of n sides passing in order through 

 the n + 1 points o^ o 2 ,....o n , o n+r Therefore (theorem 9) the 

 problem of the inscription of the closed (n + 1) 'gons whose sides 

 pass in order through the n + 1 points is partially porismatic. 

 Moreover, we know that the line xx is a closing chord of an 

 inscribable (n + l)'gon, and must, therefore, pass through the 

 pole of the plane whose trace is the locus of first extremities of 

 the inscribable closed (n + l)'gons. Hence we have the 

 following theorems : 



