44 Prof. Miller on the employment of the Gnomonic 



G, P, a second equation is obtained of the same form as the pre- 

 ceding. From these two equations the ratios of u, v, w may be 

 found. 



When P is in one of the lines 

 through two of the points D, E, F, G, 

 the equation (97), which expresses this 

 condition, may be used in the place of 

 one of the preceding equations invol- 

 ving M, V, w. 



15. Let the points D, E, F, G, no 

 three of which are in one straight line, 

 be the gnomonic projections of four 

 poles the symbols of which are known. 

 To findP, ha\Ting given its symbol uvw. 



Since u, v, w, the indices of P, are given, as well as the indices 

 of 1)E, DG, F, X, the numerical value of sin EDF sin GDP : sin 

 GDF sin EDP may be found. Hence DE, DF, DG being given, 

 and the value of X, DP may be constructed. In the same manner, 

 with ED, EG, EF, and the numerical value of sin DEG sin FEP : 

 sin FEG sin DEP, EP may be constructed. The intersection of 

 the lines DP, EP is the point P required. 



When the numerical values of u, v, w are such as to show by 

 {0) that P is in one of the lines joining every two of the points 

 D, E, F, G, one only of the lines DP, EP suffices to deter- 

 mine P. 



Or, since this line meets one of the other lines through every 

 two of the points D, E, F, G, in a third point the symbol of 

 which can be found, P may be determined by means of these 

 three points, and the numerical value of the anharmonic ratio of 

 these three points and P. 



16. Hence, if the gnomonic projections of four poles of a cry- 

 stal, no three of which are in one straight line, and their symbols 

 be given, the symbols of all the other poles can be found ; or, 

 the symbols of the latter being given, the poles themselves can 

 be determined, without any knowledge of the centre of the pro- 

 jection, or of its distance from the centre of the sphere, or of 

 any of the angles of the crystal, or even of the system of cry- 

 stallization to which it belongs. 



17. Queustedt indicates the positions of the faces of a crystal 

 by the lines in which planes drawn parallel to them, through a 

 given point 0, intersect a given plane. The lines which repre- 

 sent the faces are therefore the gnomonic projections of great 

 circles of a sphere having their planes parallel to the respective 

 faces of the crystal. The axis of a zone is represented by the 

 gnomonic projection of one extremity of a diameter of the sphere 

 drawn parallel to the axis of the zone. The lines and points 



