﻿F. 
  E. 
  Wright 
  — 
  Methods 
  in 
  Microscopical 
  Petrography. 
  537 
  

  

  Thus 
  cot 
  X 
  2 
  = 
  cos 
  37° 
  cot 
  41° 
  = 
  sin 
  (90° 
  - 
  37°) 
  cot 
  41°. 
  

   With 
  these 
  values 
  we 
  find 
  (Plate 
  YIII) 
  X 
  2 
  = 
  17°'3. 
  

   From 
  equation 
  16 
  we 
  have 
  

  

  sin 
  ^ 
  = 
  sin 
  37° 
  cos 
  41° 
  = 
  sin 
  37° 
  sin 
  (90° 
  — 
  41°) 
  

  

  Solving 
  this 
  equation 
  by 
  Plate 
  IX 
  we 
  obtain 
  

  

  /* 
  2 
  = 
  27° 
  

  

  With 
  Plates 
  YIII 
  and 
  IX 
  it 
  is 
  thus 
  possible 
  to 
  pass 
  directly 
  

   from 
  Xi,/*! 
  to 
  X 
  2 
  , 
  /jl 
  2 
  or 
  to 
  c/>, 
  p 
  without 
  any 
  computation. 
  The 
  

   above 
  examples 
  are 
  sufficient 
  to 
  indicate 
  the 
  mode 
  of 
  solving 
  the 
  

   general 
  equations. 
  They 
  do 
  not, 
  however, 
  convey 
  an 
  adequate 
  

   idea 
  of 
  the 
  wide 
  range 
  of 
  application 
  which 
  these 
  plates 
  have 
  

   in 
  optical 
  and 
  crystallographical 
  work, 
  especially 
  for 
  verifying 
  

   computations 
  and 
  the 
  values 
  obtained 
  from 
  projection 
  plots 
  

   by 
  other 
  graphical 
  methods. 
  

  

  It 
  is 
  important 
  to 
  emphasize 
  the 
  fact 
  that 
  all 
  the 
  transfor- 
  

   mation 
  equations 
  7 
  to 
  18 
  can 
  be 
  solved 
  directly 
  by 
  means 
  of 
  

   Plates 
  YIII 
  and 
  IX, 
  provided 
  proper 
  care 
  be 
  taken 
  to 
  use 
  the 
  

   complement 
  of 
  the 
  angles 
  wherever 
  necessary. 
  The 
  actual 
  

   subtraction 
  need 
  not 
  be 
  performed, 
  however, 
  as 
  the 
  comple- 
  

   ments 
  of 
  all 
  angles 
  are 
  given 
  below 
  and 
  to 
  the 
  right 
  of 
  the 
  

   actual 
  angles 
  in 
  the 
  two 
  plates. 
  

  

  Projections. 
  

  

  In 
  actual 
  practice 
  nearly 
  all 
  the 
  optical 
  properties 
  of 
  a 
  min- 
  

   eral 
  can 
  be 
  deduced, 
  if 
  the 
  shape 
  and 
  position 
  of 
  its 
  index 
  ellip- 
  

   soid 
  be 
  known 
  for 
  each 
  given 
  wave 
  length. 
  This 
  index 
  ellipsoid 
  

   has 
  certain 
  properties 
  which 
  enable 
  the 
  observer 
  to 
  determine 
  

   the 
  vibration 
  directions 
  and 
  the 
  refractive 
  indices, 
  a! 
  and 
  y\ 
  of 
  

   any 
  crystal 
  section 
  ; 
  also 
  to 
  ascertain 
  the 
  positions 
  of 
  the 
  two 
  

   optic 
  axes 
  and 
  the 
  angle 
  between 
  them. 
  By 
  virtue 
  of 
  

   these 
  properties 
  the 
  observer 
  is 
  able 
  to 
  substitute 
  in 
  place 
  of 
  

   the 
  several 
  index 
  ellipsoids 
  a 
  single 
  sphere 
  and 
  to 
  operate 
  with 
  

   that 
  alone. 
  This 
  sphere 
  in 
  turn 
  can 
  be 
  projected 
  and 
  the 
  rela- 
  

   tions, 
  which 
  obtain 
  on 
  it, 
  can 
  be 
  more 
  or 
  less 
  perfectly 
  repre- 
  

   sented 
  on 
  a 
  single 
  plane 
  (the 
  plane 
  of 
  projection). 
  Several 
  

   different 
  projections 
  are 
  in 
  use 
  in 
  optical 
  work 
  at 
  the 
  present 
  

   time, 
  each 
  of 
  which 
  has 
  its 
  advantages 
  and. 
  its 
  weak 
  points. 
  

   The 
  orthographic 
  projection 
  is 
  used 
  chiefly 
  to 
  represent 
  the 
  

   relations 
  which 
  exist 
  in 
  interference 
  figures, 
  since 
  the 
  interfer- 
  

   ence 
  figures, 
  as 
  they 
  are 
  observed 
  under 
  the 
  microscope, 
  are 
  

   orthographic 
  projections 
  of 
  the 
  interference 
  phenomena 
  in 
  space. 
  

   For 
  graphical 
  solutions 
  of 
  optical 
  problems, 
  the 
  stereographic 
  

   projection 
  is 
  commonly 
  used, 
  the 
  stereographic 
  projection 
  plots 
  

   of 
  renfield 
  and 
  Wulff 
  rendering 
  its 
  application 
  direct 
  and 
  

   accurate. 
  The 
  stereographic 
  projection, 
  however, 
  distorts 
  the 
  

   hemisphere 
  considerably, 
  the 
  length 
  of 
  an 
  arc 
  of 
  1° 
  at 
  the 
  mar- 
  

  

  