368 A. C. LANE — THE ANGLES OF CRYSTALS. 



(where Sy is an arc of a circle and SO a straight line tangent to each other 

 at S). Then we describe a circle upon 0^3, which will include the given 

 angle (BOS13P). The intersections of the circles E0Sf3 and RLSM, viz., 

 R and S, give a double solution for the vertex sought. 



All these constructions are much fticilitated by a paper scale of tangents 

 and long chords to the radius of the primitive as unity. Such a scale is 

 represented in figure 5. By its use we may rapidly construct the spherical 

 projection (as in figures 2, 3, and 4) representing any observed section, and 

 obtain numerical results by solving the spherical triangles, if the position of 

 the section can be fixed. 



Location of a kandom Section. 

 Solution of the Problem with Aid of cm Axial Image. 



§ 3. This general problem may sometimes be roughly solved with the help 

 of convergent light, if we can see a bisectrix, as in figure 4a, since the 

 position in which the hyperbolas of a biaxial image close into a cross is that 

 in which the axial plane is parallel to one of the principal sections of the 

 Nicols. Therefore, the direction of the trace of /3 normal to the axial plane 

 will be given. We can also note the direction in which the bisectrix (say y) 

 emerges. Thus we get the angle fSSy, (figure 4). We can also estimate the 

 apparent obliquity of emergence of the bisectrix, and reduce it to its true 

 value in the same manner as that in which we find 2V. This will give us 

 jS. Then we can construct figure 4 as previously described, and can deter- 

 mine the positions of S and R. 



Which of the two solutions of the general problem is. the true one may 

 commonly be determined by something else, e. g., pleochroism, dispersion, or 

 form. 



Obviously this method cannot be applied when the axial image is entirely 

 outside the field of view ; yet, so long as even one axis appears and the 

 direction of the axial plane may be inferred, we may apply it, although 

 with increasing inaccuracy as we see less of the axial image. 



In uniaxial crystals we know that a direction of extinction is parallel to 

 the basis. It has the same value, therefore, as the trace of one face. 

 The determination of the obliquity of the axis will only limit aS^ to a 

 certain circle. 



Solution for three traced Faces. 



§ 4. Another method of locating the section is by the traces of three faces 

 or crystal planes that we think we recognize. The solution of this problem 

 is the spherical counterpart of the three-point problem in trigonometry. In 



