108 G-EOMETKICAL EESEARCHES, 



THEOREM 5. 

 If in a surface of the second degree there be inscribed four 



whose sides pass in order through n fixed points ; then, representing 

 tangent planes by capital letters of like names and subscripts with 

 the small letters indicating the points of contact, we have 



5. If in the equations of the last theorem we suppose a , b 

 and d n ^ to be respectively coincident with a , b v d ; and that 

 c x and c are distinct. Then, evidently 



!il^l-. Jl^l and "i' AI . C "+i' A i 



" ' " 



From these equations we at once perceive that the closing 

 chord Cj c passes through the common point of intersection of 

 the planes A X , B , D . 



Hence, from this and theorem 4, we infer the following : 



THEOREM 6. 



If in a surface of the second degree there can be inscribed 3 

 closed rigons, whose sides pass in order through n fixed points ; 

 then according as any other inscribed rfgon having its sides passing 

 in order through the points and not having its first extremity in 

 plane with those of the others, is an open u'gon or a closed n'gon, so 

 accordingly will the entire locus of all the answerable positions for 

 first extremities of inscribable closed n'gons (whose sides pass in 



