BY MARTIN GARDINER, C.E. 113 



9. And from theorems 3 and 15 we infer 



THEOREM 17. 



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

 any odd number k ; and if in the surface there can be inscribed one 

 closed k.n'gon whose k series of successive n sides pass in order 

 through the n fixed points so that no two of the series are co-incident ; 

 then will the first points of the 1 st , n + 1 th , 2 n + 1 th , .... (k 1) 

 n + 1 th , sides lie all in the trace of one plane ; and any point in 

 this trace will be an answerable position for the first extremity of an 

 inscribable closed k.n'gon whose sides pass in order through the n 

 fixed points. 



10. The following theorem (of which theorem 9 may be 

 regarded as the particular case in which k = 2) is evident. 



THEOREM 18. 



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

 any even number k ; and if in the surface there can be inscribed a 

 closed k.n'gon whose k series of successive sides pass in order 

 through the n fixed points so that no two of the series of sides are 

 co-incident ; then will the first points of the 1 st , n + 1 th , 2 n + 1 th 

 .... (k 1) n + 1 th sides lie all in the trace of one plane ; and any 

 point in the surface will be an answerable position for the first 

 extremity of another inscribable closed k.n'gon whose sides pass in 

 like manner through the n fixed points; and the problem of the 

 inscription of the closed ^.k.n'gons whose ^.k series of sides pass in 

 order through the n fixed points is partially porismatic. 



11. Now let us have a surface S of the second degree and the 

 first n 2 points o 1? o g , .... o n _ of a series of n points, such as 

 to render porismatic the problem of the inscription of the closed 

 (n 2)'gons whose sides pass in order through the n 2 given 

 points. 



First, it may be observed that when the inscription of the 

 closed (11 2)'gons is fully porismatic, then no distinct fixed 

 positions can be found for the n 1 th and 7i th points which will 

 render porismatic the inscription of the closed tt'gons whose sides 

 pass in order through the n points of the series. But it is 



