BY MARTIN GARDINER, C.E. 109 



order through the n points) be the trace of the plane containing the 

 first extremities of the 3 closed n'gons or the entire surface in other 

 words so accordingly will the problem of the inscription of the 

 closed rfgons be partially porismatic or fully porismatic. 



THEOREM 7. 



If the problem of the inscription of closed n'gons (whose sides 

 pass in order through n fixed points) in a surface of the second 

 degree be partially porismatic, then will the closing chords of all the 

 inscribable open logons (whose sides pass in order through the n 

 points) pass through the pole of the plane whose trace is the locus of 

 the first extremities of the inscribable closed n'gons. 



THEOREM 8. 



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

 as to render partially porismatic the problem of the inscription of the 

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

 any point whatever in the surface be an answerable position for the 

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

 and whose second n sides pass in order through the n fixed points. 



6. Suppose we have a surface of the second degree and n fixed 

 points, such that in addition to two inscribed closed n'gons whose 

 sides pass in order through the points, we have an inscribed 

 closed 2 ?i'gon whose two successive series of n sides pass in order 

 through the n points. And let a l and b 1 be the first extremities 

 of the closed n'gons, and c l the first extremity of the closed 2 n'gon. 



Tben d l and d. l being the extremities of any inscribed open 

 ft'gon whose sides pass in order through the n points, we have 

 (from equations of theroem 5) 



c v A i c n+r A i _ d v A i d n+v A i 



C l> B l C +l' B l d V B J rfn+l' B i 



And as we can interchange the extremities c^ c n 1? it is evident 

 that 



, and that = And from 



