26 



E. J. CHAPMAN ON THE 



101° 40'. The other pyramids comprise : §P, with front and side angles of 149' 34' and 

 120° 6'; and 2P, with these angles equal to 130° 22' and Y4° 54'. Another triaxial pyra- 

 mid, |P2, in the vertical zone of V2, is also of occasional occurrence, but its planes as a 

 rule are very small or more or less linear. The angle over a front or macro-polar edge 

 equals 122° 51', and that over a side edge, 126° 10'. 



The Side-Polars ( = Brachy-Polars or Brachydomes) comprise : 2P, the side polar of 

 most common occurrence, with angle across the summit equal to 92° 42'; and the less 

 acute side-polars, fP with summit-angle of 108° 51', and P (a rare form) with summit- 

 angle of 129°.' Also the more acute forms, 3P (of very rare occurrence) measuring 69° 54' 

 over the summit, and 4P with summit-angle of 55° 20'. It is this latter form, in a largely 

 developed condition, which characterizes essentially the Mexican type of Topaz crystals. 



The interfacial angles in Topaz fluctuate, it is well known, within slight limits, and 

 they are not always easily obtained in consequence of the striation or want of lustre of 

 many planes. But, taking the commonly accepted angles, the axial ratios of the protaxial 

 forms are as follows : x (vertical axis) 4769*75 ; x (macrodiagoual) 1 or unity ; x (brachy- 

 diagonal) 528542. 



The symbols corresponding to those employed above, in the notations of Naumann, 

 Dana, and Miiler-Grrassmann, are shown in the following table." 



' Since this paper was drawn up, tlie writer has observed anotlier obtuse bracliydome or side-polar form, fP 

 in a colourless crystal from Japan. The summit-angle of this form, measured by application-goniometer (the 

 planes being devoid of lustre), equals llii" to 119', or by calculation, 118° 24'. The known side-polars or brachy- 

 domes, therefore, comprise the series P, IJP, IJP, 2P, 3P (a rare form) and 4P— or in the Miller-Grassmann 

 notation: Oil, 045, 023, 012, 013, 014.— E.. I. C, November, 1892. 



'See also, as regards a fuller explanation of the author's system, the Appendix to this paper. 



'In this symbol, 041, as well as in several other symbols of llie Miller-Grassmann notation, a very erroneous 

 conception is likely to be occasioned : namely, that the form, in place of being (as it really is) a very acute form, is 



