1868.] 



Uniquadric Homographics. 



395 



Third method. — Draw a^a^^ bj)^ the closing chords of the open Tj'gons 

 composing the closed 2/i'gons. If these two lines intersect in a point p, 

 draw the line ccx through the points in a^a^y bj)^ which are harmonic con- 

 jugates to p in respect to the segments a^a^, b^b.^ ; find the line ii reciprocal 

 to xx. Then will and ii pierce the quadric in the four points which are 

 first extremities of closed w'gons. But if a^a^, bj)^ do not intersect, then 

 find the final extremity c.^ of the n^oii having the point as first extremity ; 

 and if the line c^c^ cuts either a^^a^ or bj)^, the lines xx and ii can be found 

 in the manner just indicated. If c^c^ do not cut either of the lines a^a^, bj)^, 

 then find the points a.^ in which a^a^ pierces the planes bfi^c^^ ^i^a^a * 

 find the points (d.^ in which bj).^ pierces the planes a-^a.^c^, a^a.^c^ ; find the 

 points A^, h.^ which divide the segments a^a^, a.^a^ harmonically ; find the 

 points k^, which divide the segments bj)^, 13^(3^ harmonically ; through 

 the two points A,, k^y cutting a^a^, b^b^ internally, draw the linexx ; through 

 the two points h^, k.,, cutting a^a^, b^b.^ externally, draw the line ii. Then 

 will XX and ii be reciprocal lines piercing the quadric in the four points 

 which are first extremities of closed 7^'gons. 



Fourth method. — Put 2^ and to represent the homographic figures in 

 which the first and final extremities of all inscribable ?i*gons are correspond- 

 ing points ; find the point which corresponds in either of the figures S, 



to the centre a^, of the quadric regarded as belonging to the other figure ; 

 find the diameter d^d.^ which contains the points a^, ; find the points 

 q which divide the segments d^d^ and a^a^ harmonically ; draw the plane 

 P which is polar to the point p which lies outside the quadric ; find the 

 point a.^ which is final extremity of a ^'gon whose first extremity is in the 

 trace of the plane P ; draw xx the diameter of the trace of P which bisects 

 a^a^ (the point will be in the trace of P) ; find the line ii reciprocal to 

 XX. Then will xx and ii pierce in first extremities of the four inscribable 

 closed Ti'gons. But if the centre of the quadric be a double point of the 

 figures 2p S^^ proceed as follows : — Inscribe any w'gon in the quadric, and 

 draw the diameter xx which bisects its closing chord. Then will the dia- 

 meter XX and its reciprocal at infinity pierce the quadric in the four points 

 which are first extremities of closed /I'gons. 



(5) When we can inscribe an open 2/i'gon the problem is always non- 

 porismatic, and we can find the lines xx, ii which pierce the quadric in first 

 extremities of the four closed w'gons by either of the four following 

 methods : — 



First method. — Put and S,^ to represent the homographic figures in 

 which the first and final extremities of all inscribable w'gons are correspond- 

 ing points. In the figures and find the points o^ and o^ which are 

 the correspondents of the centre o of the quadric regarded as belonging to 

 the figures and S^^ ; draw the diameter ror which bisects o^o^ ; find the 

 line r^o^r^ in 2^ which corresponds to ror in S^^ ; bind the line r'r reciprocal 

 to r^r^ ; through the centre o draw the diametral plane K which bisects all 

 chords parallel to ror ; find the points pp p' in which the reciprocal lines r/^ 



