432 Proceedings of the Eoyal Physical Society. 



along the face of the cube are cat at a fractional length 

 of the Tetrad axes, whose numerator is the second figure of 

 the index and whose denominator is the first; while the 

 Triad axes are cut at a fractional length from o, whose 

 numerator is the first figure of the index and whose denom- 

 inator is the sum of the first and second. 



The last hemihedral form of the Cubic System to be 

 noticed is the Diploid, which is called also the Dyakis 

 Dodecahedron, and the Trapezoidal Icositetrahedron, etc. 

 It is the hemihedral form of the Six-faced Octahedron. It 

 is represented by both positive and negative forms, denoted 

 as usual by a /< or tt prefixed to the index of the funda- 

 mental form. 



Dana recognises 30 positive forms, ranging from (k 721) 

 to (k 543). This form is bounded by 24 irregular trapeziums 

 having only two sides equal. It has six four-faced solid 

 angles which terniinate the opposite extremities of the 

 Tetrad axes, and touch the centre of each face of the cir- 

 cumscribing cube. It has also 12 four-faced solid angles, 

 which do not lie in the Tetrad, Triad, or Dyad axes of the 

 cube. 



The Tetrad axes join the opposite four-faced solid angles, 

 while the Triad axes join the six -faced solid angles at 

 distances from o equal to those of the fundamental form 

 from which the Diploid is derived. 



In adapting the Standard Axial Cross to the other systems, 

 the axes have to be varied in length, and the neat methods 

 which are applicable to drawing most of the simple forms 

 belonging to the Cubic System are no longer available, or 

 else become of too complicated a nature to make it worth one's 

 while to employ them. For the delineation of these, there- 

 fore, recourse must be had to the methods described in more 

 detail above, in connection with the Linear Projection. 



The usual plan, after the preliminary freehand sketch of a 

 crystal has been made — as nearly as possible in all respects 

 with the same relative proportion of the faces seen in the 

 crystal so drawn — is to redraw it first on a larger scale, taking 



