GEOMETRICAL RESEARCHES, 



evident that answerable coincident positions have unlimited 

 space as locus. 



When the problem of the inscription of the closed (n 2) 

 'gons is partially porismatic, we know that the closing chords of 

 inscribed (n 2) 'gons will pass through the point x which is 

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

 extremities of the inscribable closed (n 2) 'gons. And it is 

 evident that x and X may be real even though the closed (n 2) 

 'gons be imaginary. 



Now if a l a 2 .... a n _ l be one of the inscribed (n 2) 'gons, it 

 is evident that in order to render porismatic the inscription of 

 the closed rc'gons we must have o and o^ in such positions that 

 by drawing a , o , to cut the surface in a , then will a o cut 



o n _i n.i rf n n 



the surface in a . This can be effected by taking o^_ l anywhere 

 in the plane X, and by then taking o anywhere in the polar line 

 of the point o^_ in respect to the trace of X. 



Hence we have the following theorem : 



THEOREM 19 (porismj. 



Given a surface of the second degree and the first n 2 points 

 of a series of n points such as to render partially porismatic the 

 problem of the inscription of the closed (n tycoons whose sides 

 pass in order through the n 2 given points : a plane X can be 

 found such that by taking the n 1 th point of the series anywhere 

 therein we can find a corresponding straight line in the same plane, 

 any point in which line being made a position for the n ih point of the 

 series will render partially porismatic the problem of the inscription 

 of the closed logons whose sides pass in order through the n points 

 of the series. 



12. In the investigation of the preceding theorem I have used a 

 theorem arrived at in the researches concerning w'gons inscribed 

 in curves of the second degree, viz. : " If there be any line of 

 the second degree and 3 points in its plane such that each one 

 has its polar line passing through the other two, then will any 

 point in the curve be an answerable position for the first extremity 



