80 GEOMETRICAL RESEARCHES, 



Vi 



And from these we at once obtain the equation : 



b 2l 



b n + l n 



bo a , o 



n n n+l n 



In the porismatic states of the data it is obvious the second 

 side of this equation must = 1, since a l & & 1 must be 

 coincidents with c^ & & . 



14. We may also remark that if a^ a^ w a x & l l & 2 



.... & 6 are two closed %'gons, and that o^ o , .... o the 

 intersections of their pairs of sides (of their first sides ; of their 

 second sides, &c.) are in one straight line ; then, drawing 

 straight lines through r and 6 p through a^ and & a , &c., to cut 



the line o l o 2 o n ; it is evident the line Q I o 2 .... o n has 



coincident with it a third closed %'gon inscribed in the n lines 

 &, &i> a o b^ &c. And therefore we must have : 



11 92 



n a i n 



or 



\ ( \ | ' \ 2 ........ 6 1 Q \ = 



\ 6 l r & 22 ........ b n J ' 



And as the third or single-line closed n'gon can be substituted 

 instead of either of these, we must have each of the two 

 factors of the first side = 1. 



