BY MARTIN GARDINER, C.E. 119 



of T but not to have either of its extremities pass through the 

 points in which xx pierces the surface. Hence we may announce 



THEOREM 27. 



If there be a surface S of the second degree and a series of n 

 fixed points such as to render non-porismatic the problem of the 

 inscription of the closed logons whose sides pass in order through 

 the n points of the series ; then will the closing chords of the inscribed 

 open logons whose sides pass in order through the fixed points be 

 tangents to a surface of the second degree having double contact with 

 the surface S in the two points which are answerable positions for 

 first extremities of the inscribable closed n'gons whose sides pass m 

 order through the n fixed points. 



THEOREM 28. 



If two surfaces of the second degree have double contact ; and 

 that from points in one of the surfaces we draw chords tangent to 

 the other surface in plane with the line xx of contact of the 

 surfaces, and of such roiatives (in respect to the plane sections in 

 this other surface made by the planes through xx containing these 

 tangents) as are indicated by any one of the chords we conceive 

 to revolve with the plane which contains it round the line xx as 

 axis, in such a manner as to be always tangent to the surface 

 but not to have either of its extremities pass through the points where 

 xx pierces the surface; then will the final points of the chords 

 belong to a figure which is homographic with a figure to which the 

 first extremities of these chords belong. 



THEOREM 29. 



If two surfaces of the second degree have double contact, and 

 that any point in the line of contact is the vertex of a cone enveloping 

 one of the surfaces, then- will the traces of this cone on the other 

 surface be plane curves. And the poles of the planes containing these 

 traces are situated in the line of contact of the surfaces. 



THEOREM 30. 



If there be a surface S of the second degree, and n entities, each 

 entity of which is either a fixed point or a conicoid having double 



