543 



shall be parallel to ab, and then to join SR, and examine in 

 what point a' this joining line crosses the given line ab. The 

 point a' thus found will be entirely independent of the assumed 

 initial point p, and will satisfy the condition required : in 

 such a manner that if, from any other assumed superficial 

 point p', we draw the chords p'aq', q'br', and the parallel 

 p's' to AB, the chord r's' shall pass through the same point 

 a'. All this follows easily from principles perfectly well 

 known. 



2, Since then for two given guide-points we may thus 

 substitute the system of a guide-star and a guide-point, it 

 follows that for three given guide-points we may substitute a 

 guide-star and two guide-points; and, therefore, by a repeti- 

 tion of the same process, may substitute anew a system of two 

 stars and one point. And so proceeding, for a system of w 

 given guide-points, through which n successive and rectilinear 

 chords of the surface are to pass, we may substitute a system 

 of w - 1 guide-stars, and of a single guide-point. The pro- 

 blem of inscribing, in a given surface of the second order, a 

 gauche polygon of n sides, which are required to pass succes- 

 sively through n given points, is, therefore, in general, redu- 

 cible, by operations with straight lines alone, to the problem 

 of inscribing in the same surface another gauche polygon, of 

 which the last side shall pass through a new fixed point, while 

 all its other (n - 1) sides shall be parallel to so many fixed 

 straight lines. And if the Jirst n sides of an inscribed poly- 

 gon of n+ I sides, ppi P2 . . . p„, be obliged to pass, in order, 

 through n given points, ai A2 . . . a„ , namely, the side or chord 

 ppi through Ai , &c., it will then be possible, in general, to 

 incribe also another polygon, pqi Q2 . . . p„, having the same 

 first and nth points, p and p„, and therefore the same final or 

 closing side p„p, but having the other n sides different, and 

 such that the n-\ first of these sides, pqi, Q1Q2, . .. Qn_2 

 Qb - 1 , shall be respectively parallel to w - 1 given right lines, 

 while the nth side Qn - 1 Pw shall pass through a fixed point b„. 



