545 



4. Since, for a system of three stars, we may substitute a 

 system of the centre and two stars, it follows that for a system 

 oi four stars we may substitute a system of the centre and 

 three stars ; or, by a repetition of the same process, may sub- 

 stitute a system of the centre, the same centre again, and two 

 stars ; that is, ultimately, a system of two stars may be sub- 

 stituted for a system oifour stars, the two employments of the 

 centre as a guide having simply neutralized each other, as 

 amounting merely to a return from n to p, after having go7ie 

 from p to the diametrically opposite point N. For five stars 

 we may therefore substitute three ; and for six stars we may 

 substitute four, or two. And so proceeding we perceive that, 

 for any proposed system of guide-stars, we may substitute two 

 stars, if the proposed number be even; or three, if that num- 

 ber be odd. And by combining this result with what was 

 found in (2), we see that for any given system of « guide-points 

 we may substitute a system of two stars and a point, if n be 

 odd ; or \in be even, then in that case we may substitute a sys- 

 tem of three stars and a point: which may again be changed, 

 by (3), to a system of the centre, two stars, and one point. 



5. Let us now consider more closely the S3'stem of two 

 guide-stars, and one guide-point; and for this purpose let us 

 conceive that the two first sides pq, and Qi Q3 of an inscribed 

 gauche quadrilateral PQ1Q2P3 are parallel to two given right 

 lines, while the third side Q2P3 is obliged to pass through a 

 fixed point B3; the first point p, and therefore also the qua- 

 drilateral itself, being in other respects variable. In the plane 

 PQi Q3 of the two first sides, which is evidently parallel to a 

 fixed plane, inscribe a chord QjS, whose direction shall be 

 conjugate to that of the fixed line 0B3, and therefore shall 

 itself also be fixed, o being still the centre of the surface; and 

 draw the chord PS. Then, in the plane inscribed quadrilateral 

 PQi QjS, the three first sides have fixed directions, and there- 

 fore, by (3), the direction of the fourth side sp is also fixed. 

 In the plane SQ2P3, which contains the given point Bg, draw 



