546 



through that point an indefinite right line B3C3, parallel to 

 SQ2; the line so drawn will have a given position, and will be 

 intersected, at some finite or infinite distance from B3, by the 

 chord SP3, which is situated in the same plane with it, namely, 

 in the plane SQa P3. But if we consider the section of the sur- 

 face, which is made by this last plane, and observe that the 

 two first sides of the triangle SQa P3 pass, by the construction, 

 through a star or point at infinity conjugate to B3, and through 

 the point B3 itself, we shall see that, in virtue of a well-known 

 and elementary principle respecting triangles in conies, the 

 third side P3S must pass through the point D3, if D3 be the pole 

 of the right line B3C3, which contains upon it the two conju- 

 gate points; this pole being taken with respect to the plane 

 section lately mentioned. If then we denote by D3E3 the in- 

 definite right line which is, with respect to the surface, the 

 polar of the fixed line B3 C3, we see that the chord SP3 must in- 

 tersect this reciprocal polar also, besides intersecting the line 

 B3C3 itself. Conversely this condition, of intersecting these 

 two fixed polars, is sufficient to enable us to draw the chord 

 SP3 when the point s has been determined, by drawing from 

 the assumed point p the chord PS parallel to a fixed right line. 

 We may then substitute, for a system of two guide-stars and 

 one guide-point, the system of one guide-star and two guide- 

 lines; these lines being (as has been seen) a pair of reciprocal 

 polars, with respect to the given surface. 



6. If, then, it be required to inscribe a polygon ppi P2 .. P^n 

 with any odd number 2w + 1 of sides, which shall pass suc- 

 cessively through the same number of given points, AiAs.. 

 A2« + i} we may begin by assuming a point p upon the given 

 surface, and drawing through the given points 2n+l successive 

 chords, which will in general conduct to a final point P2« + i, 

 distinct from the assumed initial point p. And then, by pro- 

 cesses of which the nature has been already explained, we can 

 find a point s such that the chord PS shall be parallel to a fixed 

 right line, or shall have a direction independent of the assumed 



