88 GEOMETEICAL RESEARCHES, 



will be an answerable position for the first angular point of a 

 straight line closed 2 w'gon, whose two successive series of n sides 

 will pass in order through the n fixed points, and whose two 

 successive series of angular points will be situated in the n fixed 

 planes. And, reciprocally, if we can form one closed 2 w'gon 

 having its first n sides distinct from its second n sides, &c. 



THEOREM 8. 



If there be 4 closed straight line rc'gons, each one of which 

 has its sides passing in order through n . fixed points, and its 

 n angular points resting on n fixed planes, and that the first 

 angular points (all in one plane P) of these closed w'gons are not 

 all in one straight line ; then will any point in the plane P be an 

 answerable position for the first angular point of another such 

 closed tt'gon. 



NOTE. 



The principal porisms (concerning plane straight line ^'gons) 

 envolved in this paper contain as particular cases all those of a 

 kindred nature, said to have been comprised in the writings of the 

 ancient Greek Geometers. 



They contain also as particular cases all those concerning poly- 

 gons which are given in the works of Professors Simsoii, Mulcahy 

 and Chasles. 



PAPER No. 3. 



Researches concerning n'gons inscribed in curves of the second 



degree, by 



MARTIN GARDINER, C.E. 



1. Let S represent any fixed curve of the second degree ; and 



let o 1? o g , o n represent n fixed points of a series, which, taken 



in order, may be designated the 1 st , 2 nd , 3 rd , and n m points 



respectively. 



