544 



The analogous reductions for polygons in conic sections have 

 long been familiar to geometers. 



3. Let us now consider the inscribed gauche quadrilateral 

 PQi Q3 Q3, of which the four corners coincide with the four 

 first points of the last-mentioned polygon. In the plane 

 Qi Q2 Q3 of the second and third sides of this gauche quadri- 

 lateral, draw a new chord Qi R2, which shall have its direction 

 conjugate to the direction of pqi, with respect to the given 

 surface. This new direction will itself be fixed, as being pa- 

 rallel to a fixed plane, and conjugate to a fixed direction, not 

 generally conjugate to that plane ; and hence in the plane in- 

 scribed quadrilateral R2Q1Q2Q3, the three first sides having 

 fixed directions, the fourth side Q3 R3 will also have its direction 

 fixed: which may be proved, either as a limiting form of the 

 theorem referred to in (I), respecting four points in one line, 

 or from principles still more elementary. And there is no diffi- 

 culty in seeing that because pqi and Qi Rg have fixed and con- 

 jugate directions, the chord PR2 is bisected by a fixed diameter 

 of the surface, whose direction is conjugate to both of their's ; or 

 in other words, that if o be the centre of the surface, and if we 

 draw thevariable diameter pon, the variable chord NRg will then 

 be parallel to the Jixed diameter just mentioned. So far, then, 

 as we only concern ourselves to construct the fourth or closing 

 side Q3 p of the gauche quadrilateral pq, Qo Q3, whose three first 

 sides have given or fixed directions, we may substitute for it ano- 

 ther gauchequadrilateralPNRjQs, inscribed in the same surface, 

 and such that while its first side pn passes through the centre 

 o, its second and third sides, NR2 and R2Q3, are parallel to 

 two fixed right lines. In other words, we may substitute, for 

 a system of three guide-stars, a system of the centre and two 

 stars, as guides for the three first sides ; or, if we choose, in- 

 stead of drawing successively three chords, pqi, q,q^, q.^Qs, 

 parallel to three given lines, we may draw a first chord PK2, 

 so as to be bisected by a given diameter, and then a second 

 chord R2 Q3, parallel to a given right line. 



