46 Prof. Miller on the employment of the Gnomonic 



three points A, D, B, the point M may be constructed by (13); 

 In like manner, the point N may be constructed from the nume- 

 rical value of AE . CN : CE . AN, and the points k, E, C. The 

 line MN then represents the face uvw. 



21. Let the symbols of three faces in one zone, represented 

 by the lines KP, KQ, KR, be given. Let the line KS re- 

 present any other face uviv in the same zone. By equating 

 the value of sin PKQ sin RKS : sin RKQ sin PKS, found by con- 

 struction, with its value as given by (v) in terms of the indices 

 of KP, KQ, KR, KS, an equation is obtained of the form 

 au-\-bv + cw = 0, vphere a, b, c are integers. The condition that 

 the face represented by KS is in the zone represented by K, [6) 

 furnishes an equation of similar form. From these two equa- 

 tions the ratios of u, v, w may be found. 



22. Let the symbols of three faces in one zone, represented by 

 the lines KP, KQ, KR, be given, and let u, v, w be the indices 

 of a face to be represented by a line KS. The numerical value 

 of sin PKQ sin RKS: sin RKQ sin PKS can be found by (v). 

 From this, and the three lines KP, KQ, KR, the line KS can 

 be constructed by the methods of art. 13. 



23. Hence, if the lines representing four faces of a crystal no 

 three of which pass through the same point, and their symbols 

 be given, the symbols of the faces represented by all the other 

 lines can be found ; or, the symbols of the latter being given, 

 the lines which represent the faces may be constructed. It is 

 not requisite that the place of the fixed point 0, or the angles 

 of the crystal, or the system of crystallization to which it belongs, 

 should be known. 



24. In the preceding investigation spherical trigonometry has 

 been employed in preference to analytical geometry of three di- 

 mensions, because a knowledge of the former is practically indis- 

 pensable to the mineralogist, while we are not bound to assume 

 that he is acquainted with the latter. Moreover, the methods of 

 spherical trigonometry appear to be more appropriate than those 

 of analytical geometry, in discussing the relations between points 

 and circles on the surface of a sphere, and the points and straight 

 lines by which they are respectively represented in the gnomonic 

 projection of the sphere. The results of arts. (6), (7), (9), (10) 

 may, however, be easily and concisely deduced by the aid of 

 analytical geometry. An expression equivalent to (6) combined 

 with (7) was obtained by Levy in this manner (Edinb. Phil. 

 Journ. vol. vi. p. 226). Its usefulness was, however, greatly im- 

 paired by the employment of a very defective notation. 



25. The plane which has for its equation 



a b c 



