516 



Prof. Miller's Crystallographic Notices. 



On the lines observed by M. Victor v. Lang on the faces of cry- 

 stals of quartz. 



It is an impoilant advantage of Neumann's graphic method 

 of indicating the positions of the faces of a crystal, that it super- 

 sedes the employment of troublesome constructions, in the solu- 

 tion of problems relating either to the dihedral angles of crystals, 

 or to the plane angles which the edges of the faces make with 

 each other. The determination of the crystallographic import 

 of the lines observed by v. Lang on the faces of the six-sided 

 pyramid, in certain crj'stals of quartz, affords a good example of 

 the superiority of this method over other methods commonly 

 used. 



According to the observations of V. Lang {Sitzungsberichte 

 der Mathem.-natunv. Classe der kais. Akademie der Wissenschaf- 

 ten, vol. XX. p. 392.), these lines occur in groups of threes, the 

 angles which two of the lines of each group make with the 

 edges in which the faces of the six-sided pyramid meet the faces 

 of the six-sided prism, being 52.° 5' and 84° 40', measured in 

 opposite directions. It is not pos- 

 sible to measure these angles with 

 much accuracy. 



Let b, y, r denote the poles of 

 211, 112, 100 respectively ; 

 re, rd zone-circles having their axes 

 parallel to the lines on the face r. 

 Then 



r6 = 38° 13', irc= 52° 5', 



hrd=^4P 40'. 



sin br = tan be cot brc. 



log sin 38° 13' 9-79144 



log tan 52 5 Q -llllO 



log tan 38 37-5 9-90254 



Hence c is the pole of 5 6 1. The correct value of cb is 

 38° 56'-8, and that of ^>rc_is 52° 34'-2. In like manner d is 

 found to be the pole of 1 5 6. The correct value of db is 81° 3'-2, 

 and that of brd is 84° 26'-2. A third line was too imperfect to 

 admit of determining its direction by observation with even mo- 

 derate accuracy. It probably makes an angle of 31° 53'-3 with 

 the edge rb, and is parallel to the axis of the zone-circle re, 

 where e is the pole of 6 I 5. e6 = 21° 3'-2. We arrive at the 

 same result supposing the lines to occur on z, a face of the 

 rhombohedrou 12 2, which forms a dirhonibohedral combina- 

 tion with 10 0. 



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