A. C. LANE — THE ANGLES OF CRYSTALS. 



location of certain points of these curves are quite easily found. In fact we 



have, if — 



a cot a ±: cot R sin (a — R) 

 fp—.-^ r — '^^ I • 



s^=90^ 



cot a -\- cot ft sin («+/?)' ' 



J cot a — cot 13 sin (« — (3) . 



~~ ~ cot a + cot [3 ~ sin (a + /?) ' 



= 45° 7-' 



cot a — cot (3 

 cot a -\- cot [3 



sin (a — /?) 



sin (« -f /3) ' 



cos 1 . 

 CSC a -\- CSC (3 sin ^ 



P) 



cot a -f cot (3 cos 1 ^^^ _^ ^^ ' 



90° t'== 



cos 

 CSC a + CSC (3 sin 

 cot a 4- cot 3 COS 



(«-/?) 



i (« + /?) 



sni 



1- (14) 



Moreover, these curves cut the axes in general at right angles while they cut 

 the primitive at an angle (3 ; so that, when the points for the above radii are 

 plotted, the curves are not difficult to sketch. 



The projection used in figure 6 gives the highest degree of symmetry to 

 the curves, and for the radii noted in (14) cp and ft enter so symmetrically 

 that solutions for one octant answer with slight modifications for other 

 octants and interchanged a and /?. 



Probable Difference between solid Angles and their Traces. 



§ 5. The probability that a given solid angle, P^CP^, will be cut to give 

 a certain plane angle is indicated in figure 6 by the area between successive 

 curves. Thus a glance at figure 6 gives us some idea of the various prob- 

 abilities. We must remember, however, that the bounding planes of a min- 

 eral cannot be cut very obliquely if they give a well-defined and easily 

 noticed outline. The same holds true of the development of cleavage cracks : 

 but it does not hold true for twinning lines, since in this case sharp inter- 

 ference bands appear along the line of juxtaposition of two twins when they 

 are placed between crossed Nicols. 



The breadth of the border (b) made by a plane (PJ depends on the thick- 

 ness of the thin section (d) and on the angle at which it is cut {SP^). The 

 formula is — 



b = d cot SPy 



