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 oi two stars may be sub- 

 stituted for a system of four 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 gone 

 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 a7iy 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 n guide-points 

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

 odd ; or if n 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 system of two 

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

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

 gauche quadrilateral pqiQ>P3 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 Qa of the two first sides, which is evidently parallel to a 

 fixed plane, inscribe a chord QoS, 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 

 PQ, Q.,s, 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 sq. P3, which contains the given point B;t, draw 



