﻿32 
  A. 
  II. 
  Phillips 
  — 
  Symmetry 
  of 
  Crystals. 
  

  

  cally 
  into 
  eight 
  similar 
  triangles, 
  representing 
  the 
  six 
  diagonal 
  

   planes 
  of 
  symmetry 
  of 
  the 
  system, 
  three 
  of 
  which 
  are 
  repre- 
  

   sented 
  in 
  an 
  octant. 
  If 
  when 
  this 
  set 
  of 
  three 
  planes 
  is 
  placed 
  

   in 
  the 
  octant, 
  one 
  of 
  the 
  six 
  triangular 
  spaces 
  is 
  lined 
  with 
  

   mirrors 
  and 
  the 
  pole 
  placed 
  within 
  it, 
  48 
  reflections 
  ma} 
  7 
  be 
  

   counted 
  or 
  the 
  hexoctahedron 
  is 
  illustrated. 
  The 
  remaining 
  

   six 
  forms 
  may 
  be 
  represented 
  by 
  placing 
  the 
  pole 
  in 
  the 
  six 
  

   possible 
  positions, 
  the 
  three 
  sides 
  and 
  the 
  three 
  angles 
  of 
  the 
  

   triangle. 
  

  

  If 
  desired, 
  the 
  disks 
  may 
  be 
  graduated 
  and 
  marked 
  off 
  in 
  

   degrees, 
  when 
  any 
  particular 
  dome 
  or 
  prism 
  may 
  be 
  illustrated. 
  

  

  The 
  model 
  is 
  also 
  most 
  convenient 
  for 
  illustrating 
  the 
  

   method 
  of 
  the 
  two-circle 
  goniometer, 
  as 
  the 
  equatorial 
  disk 
  

   represents 
  the 
  vertical 
  circle 
  of 
  the 
  instrument 
  and 
  the 
  vertical 
  

   disk 
  of 
  the 
  model 
  the 
  meridian 
  of 
  reference. 
  The 
  relation 
  of 
  

   the 
  two 
  angles 
  measured 
  on 
  the 
  instrument, 
  to 
  the 
  pole 
  of 
  the 
  

   face, 
  is 
  simply 
  shown 
  in 
  the 
  model, 
  as 
  one 
  corresponds 
  to 
  the 
  

   longitude 
  and 
  the 
  other 
  to 
  the 
  complement 
  of 
  the 
  latitude. 
  

  

  In 
  the 
  gnomonic 
  projection 
  the 
  plane 
  of 
  projection 
  would 
  

   be 
  tangent 
  to 
  the 
  two 
  vertical 
  disks 
  of 
  the 
  model 
  at 
  c, 
  and 
  the 
  

   intersections 
  of 
  these 
  two 
  disks 
  with 
  this 
  plane 
  would 
  be 
  the 
  

   lines 
  along 
  which 
  the 
  two 
  angular 
  coordinates 
  ,t 
  and 
  ^ 
  of 
  any 
  

   face 
  is 
  measured 
  ; 
  the 
  face 
  being 
  represented 
  by 
  the 
  point 
  of 
  

   intersection 
  of 
  the 
  pole 
  with 
  this 
  tangent 
  plane. 
  The 
  trig- 
  

   onometrical 
  relations 
  of 
  the 
  coordinates 
  x 
  and 
  y, 
  the 
  point 
  

   representing 
  the 
  face 
  and 
  the 
  two 
  angles 
  measured 
  on 
  the 
  

   goniometer, 
  cannot 
  be 
  more 
  simply 
  demonstrated 
  than 
  by 
  this 
  

   model. 
  

  

  Princeton 
  University, 
  May 
  3, 
  1913. 
  

  

  