﻿H. 
  E. 
  Merwin 
  — 
  Interpolations 
  on 
  Spectrograms. 
  55 
  

  

  Therefore 
  d 
  = 
  A 
  — 
  — 
  — 
  - 
  \/n 
  2 
  — 
  smH 
  — 
  sin 
  i 
  cot 
  A) 
  (3) 
  

   sin 
  (3 
  

  

  A' 
  

  

  wliich 
  may 
  be 
  written 
  d 
  = 
  —. 
  — 
  - 
  (Vn** 
  — 
  -B 
  — 
  0) 
  (4) 
  

  

  There 
  are 
  three 
  conditions 
  which 
  obtain 
  in 
  the 
  practical 
  

   application 
  of 
  equation 
  4 
  to 
  spectrograms 
  : 
  first, 
  (3 
  is 
  not 
  far 
  

   from 
  a 
  right 
  angle 
  (80° 
  to 
  100°) 
  ; 
  second, 
  n 
  2 
  is 
  between 
  about 
  

   2*4 
  and 
  3*0 
  ; 
  third, 
  the 
  ideal 
  relations 
  assumed 
  in 
  obtaining 
  

   equation 
  4 
  are 
  not 
  significantly 
  violated 
  by 
  the 
  optical 
  system 
  

   of 
  a 
  spectrograph. 
  Then 
  we 
  may 
  write 
  as 
  a 
  first 
  approxima- 
  

   tion* 
  

  

  sin 
  = 
  1, 
  and 
  yV 
  - 
  B 
  = 
  Fn 
  2 
  - 
  G 
  

  

  which 
  gives 
  d 
  = 
  A'(Fn 
  2 
  - 
  G 
  - 
  C) 
  

  

  or 
  d 
  = 
  Hn 
  2 
  - 
  K 
  (5) 
  

  

  The 
  relation 
  between 
  refractive 
  index 
  and 
  wave-length 
  for 
  

   wave-lengths 
  that 
  are 
  freely 
  transmitted 
  is 
  given 
  by 
  the 
  equa- 
  

   tion 
  

  

  n 
  2 
  = 
  a 
  + 
  e\ 
  2 
  (6) 
  

  

  Then 
  from 
  5 
  and 
  6 
  d— 
  - 
  R 
  + 
  !?(—— 
  e\ 
  2 
  \ 
  (7) 
  

  

  The 
  quantity 
  in 
  parenthesis, 
  call 
  it 
  n\ 
  can 
  be 
  evaluated 
  and 
  

   tabulated 
  for 
  various 
  values 
  of 
  \ 
  by 
  taking 
  the 
  constants 
  

   already 
  known. 
  

  

  Then 
  we 
  may 
  write 
  d 
  = 
  Kn' 
  — 
  R 
  (8) 
  

  

  That 
  is, 
  d 
  ?l 
  = 
  Kn' 
  % 
  - 
  R 
  (8a) 
  

  

  However, 
  as 
  shown 
  later, 
  not 
  the 
  exact 
  values 
  of 
  o, 
  e 
  and 
  6, 
  

   but 
  such 
  approximate 
  values 
  as 
  are 
  best 
  adapted 
  to 
  the 
  easy 
  cal- 
  

   culation 
  of 
  values 
  of 
  n' 
  need 
  be 
  used. 
  "For 
  quartz 
  each 
  of 
  

   these 
  constants 
  is 
  nearly 
  # 
  01. 
  This 
  value 
  was 
  used 
  in 
  com- 
  

   puting 
  the 
  accompanying 
  table 
  by 
  means 
  of 
  Barlow's 
  tables 
  

   of 
  squares 
  and 
  reciprocals. 
  These 
  values 
  of 
  n' 
  are 
  so 
  near 
  

   those 
  for 
  flint 
  glass 
  that 
  the 
  tables 
  may 
  be 
  used 
  in 
  interpolating 
  

   spectrograms 
  from 
  glass 
  prisms. 
  

  

  In 
  most 
  cases 
  interpolations 
  made 
  by 
  using 
  formula 
  8 
  in 
  con- 
  

   nection 
  with 
  the 
  tabulated 
  values 
  of 
  X 
  and 
  n' 
  will 
  be 
  as 
  exact 
  

   as 
  desired, 
  especially 
  if 
  one 
  or 
  more 
  comparison 
  lines 
  besides 
  

   the 
  two 
  required 
  for 
  the 
  determination 
  of 
  K 
  and 
  R 
  are 
  used 
  

   to 
  determine 
  a 
  deviation 
  curve. 
  

  

  * 
  h 
  2 
  — 
  B 
  is 
  very 
  nearly 
  equal 
  to 
  Ln 
  — 
  M, 
  so 
  that 
  an 
  equation, 
  6a, 
  may 
  be 
  

   written 
  d 
  = 
  Nn 
  — 
  P. 
  The 
  error 
  introduced 
  by 
  assuming 
  sin/3 
  = 
  1 
  is 
  much 
  

   larger 
  than 
  the 
  error 
  from 
  either 
  of 
  the 
  other 
  assumptions. 
  

  

  