﻿530 
  F. 
  E. 
  Wright 
  — 
  Methods 
  in 
  Microscopical 
  Petrography. 
  

   and 
  that 
  a 
  is 
  the 
  acute 
  bisectrix 
  ; 
  in 
  case 
  — 
  ^> 
  — 
  * 
  

  

  P 
  7 
  aft* 
  

  

  7 
  is 
  the 
  acute 
  bisectrix 
  and 
  the 
  formula 
  should 
  read 
  

  

  tan 
  2 
  Vy 
  

  

  1 
  1 
  

  

  1? 
  w 
  

  

  1 
  1 
  

  

  The 
  corresponding 
  approximate 
  equations 
  are 
  

  

  tan 
  2 
  Va 
  = 
  4r 
  J 
  ^ 
  j8 
  - 
  ! 
  and 
  tan 
  3 
  Fy 
  = 
  -- 
  JZ 
  -^r 
  

   fi 
  — 
  a 
  y 
  — 
  (5 
  

  

  To 
  solve 
  the 
  equation 
  for 
  tan 
  2 
  Va 
  pass 
  up 
  along 
  the 
  ordinate 
  

   which 
  intersects 
  the 
  abscissa 
  — 
  ^ 
  ^-, 
  respectively 
  (/3 
  — 
  a), 
  

  

  to 
  the 
  value— 
  — 
  r 
  ' 
  respectively 
  (y 
  — 
  ft). 
  The 
  diagonal 
  

  

  line 
  passing 
  through 
  the 
  point 
  thus 
  obtained 
  is 
  then 
  the 
  desired 
  

   angle 
  V. 
  (Small 
  figure 
  a, 
  Plate 
  VI.) 
  

  

  1 
  1 
  

  

  It 
  is 
  of 
  interest 
  to 
  note 
  that 
  the 
  ratio 
  — 
  — 
  is 
  always 
  

  

  smaller 
  than 
  the 
  ratio 
  — 
  - 
  

  

  t 
  - 
  p 
  

  

  a 
  2 
  

   1 
  1 
  

  

  i 
  for 
  

  

  /3 
  2 
  

   a/3 
  + 
  ay 
  y 
  — 
  ft 
  

  

  ? 
  - 
  a 
  

  

  10 
  1 
  1 
  

   a 
  2 
  /3 
  2 
  

  

  y/3 
  + 
  ay 
  /3— 
  a 
  

  

  , 
  aj8 
  +ay 
  ^ 
  

  

  and 
  ^ 
  <i. 
  

  

  y/J 
  + 
  ay 
  

  

  The 
  effect 
  of 
  substituting 
  the 
  simple 
  relation 
  

  

  1 
  1 
  

  

  tan 
  2 
  Fa 
  — 
  --—— 
  for 
  the 
  exact 
  expression 
  tan 
  2 
  Va 
  = 
  — 
  

  

  1 
  

  

  a 
  p 
  

  

  is, 
  therefore, 
  to 
  increase 
  the 
  angle 
  Va 
  and 
  to 
  obtain 
  an 
  optic 
  

   axial 
  angle, 
  2 
  Va, 
  which 
  is 
  too 
  large. 
  The 
  error 
  may 
  amount 
  to 
  

   2° 
  and 
  over 
  in 
  unfavorable 
  instances. 
  In 
  the 
  case 
  of 
  a 
  mineral 
  

  

  in 
  which 
  tan 
  3 
  Va 
  =-^ 
  is 
  slightly 
  >1, 
  2 
  V, 
  therefore, 
  nearly 
  

  

  90° 
  and 
  the 
  mineral 
  apparently 
  optically 
  +, 
  it 
  may 
  happen 
  that 
  

  

  