118 GEOMETRICAL RESEARCHES, 



such that we cannot interchange them in the divisions, we know 

 that the closing chords of these w'gons are tangents of like or 

 unlike rotatives to a conic k having double contact with the trace 

 in the points where xx pierces it (reckoning from the final ex- 

 tremities of the w'gons) just according as the final extremities 

 of the w'gons are on the same side or on opposite sides of the 

 straight line xx. Similar remarks evidently apply in respect to 

 the traces of all other planes drawn through the line xx. And 

 we know that the conies to which the closing chords are tan- 

 gents belong to a surface having double contact with S in the 

 points where xx pierces it. 



Again by assuming o , in any particular position in xx, it 

 is evident that the tangent lines from this point to the various 

 conic sections, made in the surface S by planes through xx, are all 

 closing chords to %'gons inscribable in the surface so that the 

 sides of each pass in order through the n fixed points o } , o^ ... o^. 

 And we perceive that we can have any number of closed (n + 1 ) 

 'gons inscribed in S so that the sides of each pass in order 



through the n -f 1 points o^ 2 , o^, o l ; we perceive also 



that the problem of the inscription of these closed (n + l)'gons 

 is partially porismatic. Hence we learn that the point o is 

 the vertex of a cone of the second degree enveloping the surface 

 which has double contact with S in the points where xx pierces 

 it. This is also evidently true for all other points in xx. There- 

 fore we infer that the envelope of the closing chords of all the 

 inscribable w'gons, whose sides pass in order through the n 

 points, is a surface T of the second degree having double contact 

 with S in the points in which the line xx pierces it. 



Moreover, it is evident that when we have the rotative of any 

 one of the closing chords (reckoning from the final extremity of 

 the w'gon) in respect to a section of T made by a plane through 

 xx, we can determine on the rotatives of all others, by conceiving 

 the plane to revolve round xx as axis and the chord to be 

 deformed so as to move tangentially to the various conic sections 



