154 Ford and Tillotson — Orthoclase Twins, 



time the corresponding points on IV may be located, it being 

 noted that IV 7 and III are the same arc. But one other form 

 remains to be transposed, the prism z. We have already V 

 and m' located and it is a simple matter with the aid of the 

 great circle protractor to determine the position of the great 

 circle on which they lie. Then a small circle about' J 7 with the 

 proper radius, h/\z = 29° 24/, determines at once by its inter- 

 sections with this arc the position of the poles of the z faces. 



The transposition of the faces from normal to twin position, 

 having been made, it is a simple matter to draw the crystal 

 figures from the projection.* It may be pointed out that if 

 it should be desired to make use of the methods of the gnomonic 

 projection for the drawing of the figures the stereographic 

 projection, as derived above, can be readily transformed into 

 a gnomonic projection by doubling the angular distance from 

 the center of the projection to each pole by the use of the 

 stereographic protractor. But from whichever projection it is 

 preferred to draw the figures, it is thought that the stereo- 

 graphic projection, with the aid of the Penfield protractors, 

 offers the simplest method for the ready transposition of the 

 poles of the faces from normal into twin positions. 



Mineralogical Laboratory of the Sheffield Scientific School, 

 Yale University, New Haven, Conn., April, 1908. 



* See Penfield, this Journal, xxi, 206, 1906. 



