Simpler Methods in Crystallography. 431 



whose numerator is the first figure of the index, and whose 

 denominator is the sum of all three. 



The Six-faced Tetrahedron, Boracitoid, Hemi-Hexocta- 

 hedron, or Hexakis-Tetrahedron, is derived from the Six- 

 faced Octahedron by the development of the faces constituting 

 four of its solid six-faced angles opposite the alternate solid 

 angles of the cube in which it is described. Like the other 

 hemihedral forms, it may be regarded as either positive or 

 negative, according to which set of faces is developed. 

 Dana gives the angles of twelve forms, which range from 

 (521) to (11.10.1); (531) has been observed in Boracite. 

 There is one series, regarded as positive, to which k is 

 prefixed, and the obverse of this regarded as negative, and 

 distinguished by a prefixed tt. The Tetrad Axes form the 

 four-faced solid angles. The Triad Axes join the obtuse four- 

 faced solid angles on one side of o, and the acute four-faced 

 solid angles on the other, at distances varying with the form. 

 In (k 531) the obtuse six-faced solid angles cut the Triad 

 axis on one side of o at 5/9, and on the opposite at 5/7. 



The Pentagonal Dodecahedron, Pyritoid, or Pyrito- 

 hedron, is a hemihedral form of the four-faced cube, and, 

 like the other hemihedral forms, is regarded as positive or 

 negative, distinguished respectively by k and tt, in accord- 

 ance with the half set of faces which is developed at the 

 expense of the other half. It is theoretically possible for 

 each Four-Faced Cube to be represented by its corresponding 

 hemihedral form, both positive and negative. Dana gives 

 32 positive forms, ranging from (k 10.1.0) to (k 11.10.0), 

 (k 210) being a common form. In this form the Tetrad 

 axes join the middle of the opposite six unequal edges. 

 The Triad axes join each of the eight three-faced solid angles, 

 at distances varying with the fundamental form. There are 

 also 12 three-faced solid angles which do not lie in any 

 of the three species of axes belonging to the cube, but 

 which lie in a face of the circumscribing cube, along a line 

 parallel to one of its edges, and at a distance from the 

 Tetrad axes varying with the form, being half the length of 

 the Tetrad axes in k or tt (210), in which also the Triad 

 axes are cut at 2/3 of their length. Generalising, the lines 



