﻿F. 
  E. 
  Wright 
  — 
  Methods 
  in 
  Microscopical 
  Petrography. 
  533 
  

  

  These 
  examples 
  and 
  the 
  above 
  discussion 
  suffice 
  to 
  show 
  how 
  

   much 
  information 
  can 
  be 
  readily 
  gathered 
  from 
  the 
  principal 
  

   refractive 
  indices 
  of 
  a 
  mineral. 
  

  

  Transformation 
  equations 
  for 
  projection 
  work. 
  — 
  In 
  both 
  

   cry 
  stall 
  ographical 
  and 
  optical 
  work 
  it 
  is 
  often 
  of 
  advantage 
  to 
  

   rotate 
  the 
  projection 
  about 
  one 
  or 
  more 
  axes 
  and 
  thus 
  to 
  shift 
  

   the 
  positions 
  of 
  all 
  directions 
  relative 
  to 
  any 
  specified 
  direction 
  

   such 
  as 
  the 
  pole 
  of 
  the 
  projection. 
  On 
  rotation 
  of 
  a 
  sphere 
  

   about 
  an 
  axis, 
  all 
  points 
  on 
  the 
  sphere 
  travel 
  along 
  circles 
  

   whose 
  planes 
  are 
  normal 
  to 
  the 
  axis 
  of 
  rotation. 
  Thus 
  if 
  we 
  

   denote 
  the 
  position 
  of 
  a 
  point 
  P 
  by 
  two 
  angles 
  X 
  n 
  and 
  /*„ 
  and 
  

   then 
  rotate 
  the 
  sphere 
  about 
  the 
  horizontal 
  axis 
  OE, 
  the 
  point 
  

   P 
  travels 
  to 
  P' 
  along 
  the 
  small 
  circle 
  PI 
  J/ 
  , 
  the 
  angle 
  /jl 
  1 
  remain- 
  

   ing 
  unchanged 
  throughout 
  the 
  rotation. 
  By 
  thus 
  expressing 
  

   the 
  positions 
  of 
  all 
  points 
  by 
  means 
  of 
  the 
  coordinate 
  angles 
  

   \ 
  and 
  /* 
  x 
  , 
  we 
  can 
  rotate 
  the 
  projection 
  about 
  the 
  horizontal 
  

  

  Fig. 
  6. 
  

  

  axis 
  by 
  simply 
  adding 
  or 
  subtracting 
  the 
  angle 
  of 
  rotation 
  

   from 
  the 
  angles 
  X 
  x 
  , 
  of 
  all 
  given 
  points, 
  the 
  angles 
  thus 
  obtained 
  

   locating 
  the 
  positions 
  of 
  all 
  points 
  after 
  the 
  rotation. 
  If 
  now 
  we 
  

   wish 
  to 
  rotate 
  the 
  projection 
  about 
  the 
  vertical 
  axis 
  ON, 
  it 
  is 
  

   necessary 
  to 
  ascertain 
  the 
  angles 
  X 
  2 
  , 
  /jl^ 
  (fig. 
  6) 
  which 
  corre- 
  

   spond 
  to 
  X 
  n 
  /Aj-of 
  the 
  first 
  position. 
  This 
  is 
  accomplished 
  by 
  

   means 
  of 
  equations 
  15 
  and 
  16 
  above, 
  which 
  can 
  be 
  solved 
  by 
  

   Plates 
  Till 
  and 
  IX. 
  

  

  Plate 
  YIII 
  is 
  based 
  on 
  equation 
  15, 
  which 
  can 
  be 
  written 
  

  

  tan 
  A„ 
  

  

  tan 
  }l 
  { 
  

  

  sin 
  (90°— 
  \) 
  

  

  (1 
  5a) 
  

  

  The 
  tangent 
  function 
  extends 
  from 
  to 
  oo 
  for 
  values 
  of 
  fi 
  1 
  

   from 
  0° 
  to 
  90°. 
  In 
  order 
  to 
  plot 
  the 
  entire 
  function 
  under 
  

  

  