APPLICATION TO TOURMALINE. 881 



which is the same thing, from the prism axis to the normal to the trace of 

 P,. The corresponding value for P^ is ?i~\ These values we may observe 

 and, plotting them, see what ellipses they approach and what rhombohedra 

 they indicate. Figure 7 shows the plotting of a number of such observa- 

 tions on tourmaline, together with the ellipses corresponding to R and ± l 

 of Dana. We may derive an easily applied rough rule from the ellipses at 

 once. We see, for example, that the greater angle from the trace of the 

 basis to that of the rhombohedron should, for the fundamental rhombohe- 

 dron, be between 42° and 47°. Allowing for variation on account of the 

 section not being exactly prismatic, and on account of mistaking edges for 

 the traces of faces, we see that R and zh 2 of Dana are doubtless the pre- 

 vailing forms. The former is more common, and terminates the "antil- 

 ogous " end, while the latter alternates with the basis on the " analogous " 

 end. 



Calcite and tourmaline are near enough in their parameters to permit 

 the application of figure 7 to distinguish between the R cleavage and — h 

 pressure twinning of calcite. 



Michigan Mining School, Houghton, Mich., Jiine 18, 1890. 



LVI— Bull. Geol. Soc. Am., Vol. 2, 1890. 



