﻿144 
  H. 
  S. 
  Uhler 
  — 
  Note 
  on 
  Paper 
  by 
  Ch. 
  Fabry. 
  

  

  faces 
  are 
  equally 
  inclined 
  to 
  the 
  collimating 
  line, 
  EF. 
  

   The 
  transparent 
  ends, 
  PR 
  and 
  QS, 
  of 
  the 
  immersion 
  

   trough 
  are 
  plane-parallel 
  sheets 
  of 
  glass 
  whose 
  planes 
  

   are 
  normal 
  to 
  the 
  line 
  EF. 
  The 
  problem 
  is 
  to 
  express 
  N 
  

   as 
  a 
  convenient 
  function 
  of 
  v, 
  n, 
  A, 
  and 
  D, 
  where 
  

  

  v 
  = 
  the 
  index 
  of 
  refraction 
  of 
  the 
  air 
  for 
  a 
  given 
  wave-length, 
  

   n 
  = 
  the 
  index 
  of 
  refraction 
  of 
  the 
  prism 
  for 
  a 
  given 
  wave-length, 
  

   N 
  ■= 
  the 
  index 
  of 
  refraction 
  of 
  the 
  liquid 
  for 
  a 
  given 
  wave-length, 
  

   A 
  = 
  the 
  refracting 
  angle 
  of 
  the 
  standard 
  prism, 
  

   D 
  = 
  the 
  total 
  deviation 
  of 
  the 
  ray 
  with 
  respect 
  to 
  EF. 
  

  

  The 
  definitions 
  of 
  the 
  angles 
  p, 
  y, 
  8, 
  and 
  e 
  may 
  be 
  in- 
  

   ferred 
  from 
  the 
  diagram. 
  By 
  plane 
  geometry 
  and 
  the 
  

   law 
  of 
  single 
  refraction 
  in 
  a 
  principal 
  plane, 
  it 
  is 
  evident 
  

   that 
  the 
  following 
  equations 
  hold 
  : 
  

  

  £+7 
  = 
  A. 
  

  

  8=e-iA. 
  

  

  n 
  sin 
  ft 
  = 
  N~ 
  sin 
  -J- 
  A 
  . 
  

  

  n 
  sin 
  y 
  = 
  iV^sin 
  e. 
  

  

  v 
  sin 
  D 
  = 
  i^sin 
  8. 
  

  

  Elimination 
  of 
  /?, 
  y, 
  3, 
  and 
  e 
  from 
  these 
  five 
  relations 
  leads 
  

   at 
  once 
  to 
  

  

  (tan 
  -M) 
  [+ 
  |/ 
  (N 
  % 
  - 
  v 
  2 
  sin 
  2 
  Z>)] 
  + 
  vsin 
  D 
  + 
  iVcos 
  A 
  tan 
  \A 
  = 
  

   2 
  (sin 
  tA) 
  [+ 
  V 
  (» 
  a 
  - 
  N* 
  sin 
  2 
  ±A)] 
  (1) 
  

  

  Equation 
  (1) 
  is 
  not 
  in 
  a 
  form 
  suitable 
  for 
  the 
  practical 
  

   calculation 
  of 
  N, 
  hence 
  the 
  analytical 
  processes 
  must 
  be 
  

   continued. 
  

  

  Eationalization 
  of 
  equation 
  (1) 
  gives 
  

  

  4^yiv 
  4 
  -j-/> 
  2 
  (vV-4vy6- 
  2 
  -4?iy) 
  jy*+zvprs 
  (*v-»y) 
  jst+ 
  

  

  vV-2i^nyr* 
  1 
  +n 
  4 
  p 
  4 
  =0 
  (2) 
  

  

  where, 
  to 
  save 
  space, 
  # 
  = 
  sin 
  A, 
  q 
  = 
  sin^^, 
  r 
  = 
  cos 
  J., 
  

   and 
  s 
  =sin 
  D. 
  The 
  algebraic 
  advantage 
  of 
  taking 
  A 
  = 
  

   90°, 
  — 
  which 
  is 
  the 
  only 
  case 
  discussed 
  by 
  Fabry, 
  — 
  is 
  that 
  

   this 
  value 
  removes 
  the 
  term 
  in 
  N 
  and 
  reduces 
  the 
  biquad- 
  

   ratic 
  to 
  a 
  simple 
  quadratic 
  in 
  N 
  2 
  . 
  

  

  To 
  obtain 
  the 
  required 
  power 
  series 
  in 
  s 
  assume 
  

  

  j\T= 
  a 
  + 
  bs 
  + 
  cs 
  2 
  + 
  «■ 
  +/s 
  4 
  + 
  gs" 
  (3) 
  

  

  where 
  a, 
  &, 
  c, 
  e, 
  f, 
  and 
  # 
  are 
  as 
  yet 
  undetermined 
  coeffi- 
  

   cients. 
  In 
  the 
  following 
  work, 
  terms 
  involving 
  s 
  to 
  de- 
  

   grees 
  higher 
  than 
  the 
  fifth 
  will 
  be 
  neglected 
  as 
  such 
  terms 
  

   have, 
  in 
  general, 
  no 
  influence 
  on 
  the 
  fifth 
  decimal 
  place 
  in 
  

   the 
  numerical 
  value 
  of 
  N. 
  

   Accordingly 
  

  

  