260 Penfield — Solution of Prohleins in Crystallography 



^ = 108° 44^, and a = 103° 18'. The lines x, -x and y, -y, 

 therefore, are not the true crystallographic a and h axes, but 

 the somewhat foreshortened orthographic projections of the 

 axes. ^Notwithstanding foreshortening, however, for all pur- 

 poses of calculation in the zone under consideration, the lines 

 x,—x and y, — y are lines of reference^ regarding which the law 

 of definite mathematical ratio of crystal faces holds true, the 

 same as for real crystallographic axes. The angle between 

 x^—x and y^ — y is determined by the measurement of h (010) 

 on a (100), and the angles of a third form, m (110), establishes 

 the relative lengths of the lines of reference Ox and Oy. Any 

 fourth face in the zone, therefore, when referred to Ox and Oy 

 must have an inclination in conformity with the fundamental 



law of definite mathematical ratio. For example, in figure 10 

 the face/" is parallel to the line xz which intersects the line of 

 reference Oy at one-third its length ; the symbol of /' is there- 

 fore 130, and the angle h,\f is wholly dependent upon the 

 two angles l)/\a and h/\in. Hence it is, that knowing the 

 symbols and angles of three faces in a zone, the angle of a 

 fourth face may be found if its symbol is given, or its symbol 

 may be determined if its angle is given. 



Figure 10 furnishes a key to the solution of problems like 

 the one just given. Let it be assumed that on a rhodonite 

 crystal two angles have been measured, as follows : h /\a — 

 94° 26' and hAm =- 45° 53', and that the poles 010, 100 and 

 110 are plotted on a divided circle, figure 11. Lines of ref- 

 erence x,—x and y^—y are then drawn through the center, 

 parallel, respectively, to the h and a faces^ therefore intersect- 



