1868.] 



Uniquadric Homographics. 



393 



centre ol^^ of the quadric ; assume On-\ as vertex and its polar plane as axis, 

 and find the point a -2 homological to a_i ; and proceed thus in reverse 

 order through the rest of the n given points until arrived at the point a-n 

 (the points ccn, oc-n will be distinct). Draw the diametral plane which 

 bisects the chords parallel to the line oc-n (it will also bisect a« oc-n) ; 

 assume any two points in the trace of this plane as first extremities, and 

 inscribe two w'gons in the quadric ; through the point a« and the final 

 extremities of these ^i'gons draw the plane A„ ; find the hne of intersection 

 ii of the planes A^, A„, and its reciprocal ocx. Then will xoj always pierce 

 in the two points (real or imaginary as may be) which are the first extremi- 

 ties of the only inscribable closed w'gons ; and the line ii will pierce in first 

 extremities of closed 2?i'gons. 



Fourth method. — Find the points cc-n and ctn as in last method ; assume 

 any point in the surface as first extremity, and inscribe a jz'gon whose 

 last extremity we may represent by an ; draw the plane which contains 

 the line cc-ncc^ and the point ; draw the plane D2 which contains the 

 a^cCfy and the point a^; in the lines oc^^oc-n, cc^ccn find the points tn^y 

 such that 



a,m, d^m^ V a, D, . D, 

 and in the same lines find the points h^, h.^ such that 



^.K V a,D,.a,D, ' 



Put to represent the homographic figures in which the first and 



final extremities of inscribable ?i'gons are corresponding points. Regard 



and h^ as points in and find their correspondents m^^ A3 in 2^^ ; draw 

 the planes m^m^n^^ ^i^2^3' their line of intersection xx and its re- 



ciprocal ii. Then will the line xx pierce the quadric in the only points 

 (real or imaginary as may be) which are first extremities of inscribable 

 closed ?i'gons, and the line ii will pierce in first extremities of closed 

 2?z'gons. 



Moreover the planes m^m^m^, ^J^Jh ^"^1 double planes of the 



figures S^, which are non-tangential to the quadric. 



N.B. When the points cL-n, oin are coincident, the inscription of the 

 closed Ti'gons is partially porismatic, and one of the two points which divide 

 a-n cLyi a^icl the diameter coincident with it harmonically is the point of con- 

 currence of the closing chords of all inscribable open ?^'gons, and the polar 

 plane of which passes through the other, &c. 



When the centre of the quadric is a double point, then according as all 

 the closing chords are parallels or pass through the centre, so will the locus 

 of the first extremities of the closed w'gons be the trace of a diametral plane 

 or of a plane at infinity. 



When the number n of given points is even. 

 Assume three points «p 6^ on the surface, no two of which are on 



