122 GEOMETRICAL RESEARCHES, 



problem to the inscription of a closed (n 2)'gon whose sides 

 pass in order through the n 2 known points h, o^ o^ ... o . 



If o , o 2 , o , be neither in one straight line nor such that the 

 polar plane of each one passes through the other two, then we 

 can find the straight line xx such that by assuming q any point 

 therein, we can find a corresponding point r, in the same line, 

 such that q a and r a will cut each other in a point p in the 

 surface S. And therefore evidently we can reduce the solution 

 of the problem to the inscription of the closed (n l)'gon 

 a pa a .... a n a whose sides pass in order through the n 1 

 known points q, r, o , o & , o^. 



So now it is evident we can reduce the solution of the 

 problem of the inscription of the closed w'gons to that of the 

 inscription of closed (n 2)'gons or to that of the inscription 

 of closed (n l)'gons. And thus, step by step, we can reduce 

 the problem until we make its solution depend on that of the 

 inscription of closed 3'gons, or 2'gons whose sides are required 

 to pass through known points. 



When we reduce the problem to the inscription of closed 

 3'gons whose sides are required to pass through 3 known points, 

 and that these points are in one straight line or that each one 

 of them has its polar plane containing the other two ; then will 

 the problem of the inscription of the closed rc'gons be partially 

 porismatic ; and the locus of the first extremities of the closed 

 w'gons is the trace of the polar plane of the point through which 

 the closing chords of the inscribable open rc'gons all pass. 



When we reduce the problem to the inscription of closed 

 2'gons whose sides are required to pass through 2 known points, 

 and that these points are co-incident, then we know that the 

 problem of the inscription of the closed ra'gons is fully porismatic. 



ll^p I need scarcely state that the method of solution just 

 indicated is complete, though it is obvious there are many peculiar 

 states of the data from which we can at once pronounce on the 

 nature of the solution without going through all the indicated 

 operations or processes. 



