548 



the centre, two stars, and one point; and in (5) we reduced 

 the system of two stars and a point to the system of a star 

 and two polars. In order then to find a point p which shall 

 coincide with the point v-in deduced from it as above, or which 

 shall be adapted to be the first corner of an inscribed polygon 

 of 2n sides passing respectively through the 2n given points, 

 Ai • . A2„, we must endeavour to find a chord PS which shall be 

 at once bisected by the fixed diameter ab, and shall also inter- 

 sect the two fixed polars above mentioned. And conversely, 

 if we can find any such chord PS, it will necessarily be at least 

 one chord of solution of the problem; understanding hereby, 

 that if we set out with either extremity, p or s, of this chord, 

 and draw from it 2ra successive chords pp,, &c., or ssj, &c., 

 through the 2n given points Aj, &c., we shall be brought back 

 hereby (as the question requires) to the point with which we 

 started. For, in a process which we have proved to admit of 

 being substituted for the process of drawing the 2n chords, we 

 shall be brought first from p to s, and then back from s to p ; 

 or else first from s to p, and then back from p to s : provided 

 that the chord of solution ps has been selected so as to satisfy 

 the conditions above assigned. 



8. To inscribe then, for example, a gauche chiliagon in an 

 ellipsoid, ppi .. P999, or ssi .. S999, under the condition that its 

 thousand successive sides shall pass successively through a 

 thousand given points Ai .. aioooj we are conducted to seek to 

 inscribe, in the same given ellipsoid, a chord ps, which shall 

 be at once bisectedby a given diameter a.b, and also crossed by 

 a given chord cd, and by the ■polar of that given chord. Now 

 in general when any two proposed right lines intersect each 

 other, their respective polars also intersect, namely, in the 

 pole of the plane of the two lines proposed. Since then the 

 sought chord Ps intersects the polar of the given chord cd, it 

 follows that the polar of the same sought chord ps must in- 

 tersect the given chord cd itself. We may therefore reduce 

 the problem to this form : to find a chord ps of the ellipsoid 



