Prof. W. H. Miller on Quartz, Ice, and Karstenite. 123 



Each of these crystals exhibits one face of a rhombohedron, 

 having angles which differ too widely from those of the forms 

 described by Des Cloizeaux in his Manuel de Mineralogie to admit 

 of identification with any of them, and therefore has probably 

 never been observed before. 



The larger of the two crystals, besides the supposed new face, 

 which will be denoted by the letter S, has the forms 2 1 I, 100, 

 122, 8ll, 1011, «142, «412. The faces s, 100 are 

 rather uneven, the bisection of the images of the bright signal 

 being uncertain to the extent of about 2' in the former and rather 

 less in the latter. Three observations of the angle between these 

 faces gave 30° 23'-5, 30° 23''5, 30° 24^ respectively. 



The_ other crystal has the forms 2 11, 1 0, 1 2 2, « 4 1 2, 

 a 4 1 2 in addition to $. This last face is very even and bright ; 

 but 1 is rather imperfect. The observed angle between these 

 faces lies between 30° 22'*2 and 30° 28H. 



Of the faces given by Des Cloizeaux, those which most nearly 

 approach the position of s" make with 10 angles of 29° 26', 

 30° 4', 30° 44', having for their symbols 114 4, 8 3 3, 13 5 5 

 respectively. In order to obtain more probable values of the 

 indices of S, let us suppose the angle between 100 and 5* to be 

 30° 24', hkk the symbol of S, and D, T the angles which the axis 

 of the rhombohedron makes with normals to the faces 10 0, 

 hkk. Then, since D = 51° 47' and T-D = 30° 24', we have 

 T = 82°ll'. 



But h — k tanT K „_.,- 



h + 2k tan D 



The converging fractions approximating to this number are : — 



5 6 17 23 86 



r r T t is 



The first two fractions give the faces 11 4 4, 13 5 5 already 

 noticed; the third a face 37 14 14, making an angle of 30° 18' 

 with 10 0, and therefore not very probable ; the fourth a face 

 making with 1 an angle of 30° 25', which, taking into ac- 

 count the imperfections of the faces of the crystal, agrees suffi- 

 ciently well with the observations. The resulting symbol is 

 50l9l9. 



40 



The fraction — , obtained by adding the numerators and de- 

 nominators of the third and fourth fractions, leads to the symbol 

 29 11 11. The face of which this is the symbol makes with 

 1 an angle of 30° 22', and is therefore hardly so probable as 



