﻿F. 
  E. 
  Wright— 
  Methods 
  in 
  Microscopical 
  Petrography. 
  531 
  

  

  1 
  1 
  

  

  the 
  ex-act 
  relation 
  tan 
  2 
  Va 
  = 
  -—— 
  y 
  is 
  < 
  1 
  and 
  the 
  mineral 
  

  

  is 
  actually 
  optically 
  — 
  , 
  even 
  though 
  the 
  acute 
  bisectrix 
  is 
  

   direction 
  of 
  stronger 
  birefringence 
  (7— 
  /3>/3— 
  a). 
  In 
  this 
  

   instance 
  the 
  ordinary 
  rule 
  that, 
  in 
  a 
  biaxial 
  mineral, 
  the 
  bire- 
  

   fringence 
  of 
  a 
  section 
  cut 
  normal 
  to 
  the 
  acute 
  bisectrix 
  is 
  less 
  

   than 
  that 
  of 
  a 
  section 
  normal 
  to 
  the 
  obtuse 
  bisectrix, 
  is 
  invalid 
  

   and 
  the 
  reverse 
  is 
  true. 
  Such 
  a 
  reversal 
  of 
  sign 
  can 
  only 
  occur 
  

   on 
  an 
  optically 
  negative 
  mineral 
  with 
  large 
  optic 
  axial 
  angle, 
  

   2 
  Va 
  approximately 
  90°. 
  The 
  general 
  rule 
  is 
  based 
  on 
  the 
  

   approximate 
  equation 
  (4) 
  above 
  and 
  is 
  valid 
  for 
  practically 
  all 
  

   rock-making 
  minerals. 
  To 
  illustrate 
  this 
  inference, 
  let 
  the 
  prin- 
  

   cipal 
  refractive 
  indices 
  of 
  a 
  mineral 
  be 
  a 
  = 
  1-511, 
  /3 
  = 
  1*634, 
  

   and 
  7 
  = 
  1-764. 
  In 
  this 
  case 
  /3 
  - 
  a= 
  0*123 
  and 
  7 
  - 
  /3= 
  0*130. 
  

  

  Q.I 
  OQ 
  

  

  From 
  the 
  approximate 
  formula 
  we 
  have 
  tan 
  2 
  V 
  y 
  — 
  from 
  

  

  vv 
  * 
  0130 
  

  

  which 
  we 
  find 
  (Plate 
  YI) 
  2 
  V 
  y 
  = 
  88*4°. 
  But 
  1/V 
  = 
  0*437997, 
  

   1//3 
  2 
  = 
  0*374538, 
  and 
  I/7 
  2 
  = 
  0*321368 
  ; 
  -L 
  - 
  \ 
  = 
  0*053270, 
  

  

  p 
  y 
  

  

  4r 
  - 
  4r 
  = 
  0-063459. 
  Hence, 
  tan 
  2 
  Va 
  = 
  °^ 
  32 
  ! 
  f 
  rom 
  which 
  

   a 
  £ 
  2 
  0-063459' 
  

  

  we 
  find 
  (Plates 
  YI 
  and 
  YII) 
  2 
  V 
  a 
  = 
  85°. 
  If 
  we 
  were 
  to 
  judge, 
  

   therefore, 
  from 
  the 
  principal 
  birefringences 
  alone 
  and 
  to 
  apply 
  

   the 
  above 
  rule, 
  we 
  would 
  consider 
  the 
  mineral 
  optically 
  4- 
  with 
  

   7, 
  the 
  acute 
  bisectrix, 
  and 
  2Vy 
  = 
  88°*4 
  from 
  the 
  approximate 
  

   equation 
  4, 
  while 
  in 
  reality 
  the 
  mineral 
  is 
  optically 
  — 
  with 
  a, 
  

   the 
  acute 
  bisectrix, 
  and 
  2 
  Va 
  = 
  85°. 
  In 
  the 
  examples 
  below 
  

   this 
  relationship 
  will 
  be 
  clearly 
  shown. 
  

  

  In 
  the 
  preparation 
  of 
  Plate 
  YI 
  the 
  following 
  short 
  table 
  III 
  

   of 
  the 
  values 
  of 
  tan 
  2 
  V 
  for 
  each 
  degree 
  from 
  0° 
  to 
  45° 
  was 
  

   found 
  useful. 
  The 
  values 
  are 
  listed 
  to 
  five 
  places 
  ; 
  they 
  were 
  

   computed, 
  however, 
  to 
  eight 
  places. 
  

  

  Examples: 
  — 
  (1) 
  What 
  is 
  the 
  optic 
  axial 
  angle 
  of 
  fayalite, 
  

   whose 
  principal 
  refractive 
  indices 
  are 
  a 
  = 
  1*824, 
  ft 
  — 
  1-864, 
  

  

  and 
  7 
  = 
  1-874 
  ? 
  From 
  Table 
  II 
  we 
  find 
  ~ 
  = 
  0-300573, 
  4r 
  = 
  

  

  a 
  /5 
  

  

  0-287812, 
  and-V 
  =0*284748. 
  Accordingly-^- 
  ^_= 
  0-003064, 
  

   and 
  -\ 
  - 
  4r 
  = 
  0*012761, 
  and 
  

  

  a 
  p 
  

  

  0-003064 
  7X0-003064 
  0*021448. 
  

  

  tan 
  3 
  Va 
  = 
  

  

  0*012761 
  " 
  7X0*012761 
  "~ 
  0'089327 
  

  

  