547 



and variable position of p ; and that the chord sp^n , i shall at 

 the same time cross two other fixed right lines, which are reci- 

 procal polars of each other. In order then to find a new point 

 p, which shall satisfy the conditions of the proposed problem, 

 or shall be such as to coincide with the point p-2n^i, deduced 

 from it as above, we see that it is necessary and sufficient to 

 oblige this sought point p to be situated at one or other ex- 

 tremity of a certain chord ps, which shall at once be parallel 

 to a fixed line, and shall also cross two fixed polars. It is 

 clear then that we need only draw two planes, containing re- 

 spectively these two polars, and parallel to the fixed direction ; 

 for the right line of intersection of these two planes will be the 

 chord of solution required ; or in other words, it will cut the 

 surface in the two (real or imaginary) points, p and s, which 

 are adapted, and are alone adapted, to be positions of the first 

 corner of the polygon to be inscribed. 



7. But if it be demanded to inscribe in the same surface a 

 polygon PPiPa .. P2n- 15 with an even number 2m of sides, pass- 

 ing successively through the same even number of given points, 

 A,A2 .. A2„, the problem then acquires a character totally dis- 

 tinct. For if, after assuming an initial point p upon the sur- 

 face, we pass, by 2n successive chords, drawn through the 

 given points ai, &c., to a final point Pgn upon the surface, 

 which will thus be in general distinct from p ; it will indeed be 

 possible to assign generally two fixed polars, across which, as 

 two given guide-lines, a certain variable chord SP2n is to be 

 drawn, like the chord sp.„. , of (6); but the chord ps will not, 

 in this question, be parallel to a given line, or directed to a 

 given star; it will, on the contrary, by (3) (4) (5), be bisected 

 by a given diameter^ which we may call ab ; or, if we prefer to 

 state the result so, it will be now the supplementary chord ns 

 of the same diametral section of the surface (n being still the 

 point of that surface opposite to p), which will have a given 

 direction, and not the chord ps itself. In fact, at the end of 

 (4), we reduced the system of 2w guide-points to a system of 



