430 Proceedings of the Royal Physical Society. 



where these maps do not suffice, the student will need to 

 project the two sets of faces in the manner already described. 



Drawing Hemihedral Forms, — The Tetrahedron is drawn 

 readily enough by connecting the extremities of one pair of 

 the alternate Triad Axes, and drawing lines to the pair on 

 the opposite side which alternate with them in position. 

 The Twelve-faced Trapezohedron, Deltoidal or Trapezo- 

 hedral Dodecahedron, which is the hemihedral form of the 

 Three-faced Octahedron, can be drawn by means of a modi- 

 fication of the method adopted for its primitive. The Tetrad 

 Axes are cut at normal lengbh. The Triad Axes are cut 

 dissimilarly on alternate opposite sides, so as to limit one 

 three-faced solid at one distance, and on one-half of the axis, 

 and a second alternating with it, at a different distance on 

 the other alternating with it. There are also positive and 

 negative forms, according to how these are placed. One-half 

 of each Triad Axis is cut at a fractional length from o, whose 

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

 nominator is the sum of all three, and the opposite half is 

 cut at a fractional length from o, whose numerator is the 

 first figure of the index, and whose denominator the sum of 

 the second and third, e.g. (k 221) 2/5 on one half and 2/3 on 

 the half opposite o ; (k 332) 3/8 on one half and 3/5 on the 

 other. 



The Three-faced Tetrahedron, or Kuproid, Triakistetra- 

 hedron, Hemitetragonal Trisoctahedron, etc., is derived from 

 the Twenty-four-faced Trapezohedron by the development 

 of one-half of its face and the suppression of the other. 

 According as one set, or the other set, is developed, the 

 crystals are regarded as positive or negative forms, dis- 

 tinguished respectively by a prefixed k or rr. Dana notices 

 18 forms ranging from (13.1.1) to (322), (211) being of 

 common occurrence. The Tetrad axes cut the middle of 

 each of the opposite longer edges. One-half of each Triad 

 axis is cut at normal length, and the opposite half is cut at 

 a lesser length dependent upon the particular form under 

 consideration. In (211) this is 2/4 = J; in (311) it is 3/5 ; 

 in (322) 3/7 from o. In general, therefore, the alternate 

 opposite Triad axis is cut at a fractional length from o, 



