120 GEOMETRICAL RESEARCHES, 



contact with S ; and if there be n'gons inscribed in the surface S so 

 that each side of every n'gon will meet with the entity of the series 

 which is of like rank in the series which such side is in the n'gon, 

 and in such a manner as to pass through the entity if it be a point 

 or, if the entity be a conicoid, to be tangent of certain prescribable rota- 

 tive to the trace made on this conicoid by a plane containing the first 

 point of such side and the line of contact of the conicoid with S : then 

 will the extremities of each n'gon be corresponding points of homo- 

 graphic figures. And when we cannot interchange the extremities of 

 any of the n'gons so as to have them still corresponding in the 

 homographic figures, then will the closing chords of the logons be tan- 

 gents of determinate rotatives to the traces made on a certain conicoid 

 having double contact with S by planes containing the final extremi- 

 ties of the n'gons and the line of contact of such conicoid with S. 

 But when we can interchange the distinct extremities of the tigons 

 so that they still remain corresponding points of the hornographic* 

 figures, then will the closing chords of the rigons all pass through 

 one point. 



19. It is evident we can extend many of the preceding 

 theorems by a substitution of such entities as those implicated in 

 the theorem just enunciated for some of the entities which we 

 considered as all composed of points. 



It is also obvious that all these theorems have " duals," which 

 are easy of formation by well known methods or which can be 

 arrived at by steps correlative to those we have employed in the 

 preceding investigations. As it would be superfluous to repeat 

 the " duals " of all the theorems, I will content myself by giving 

 that of theorem 20 in order to exhibit a sort of nomenclature 

 which may be found convenient in enunciating the duals of the 

 others. 



THEOREM 31. 



If there be a surface of the second degree and four fixed planes 

 such that each plane is the polar of the point common to the other 

 three planes ; then any straight line m any of the planes is an 

 answerable position for an angular joint of a closed planes u'gon 

 whose n planes are tangential to the surface, and whose four succes- 

 sive angular joints lie in the four fixed planes taken in any order 

 whatever. 



