﻿56 
  H. 
  E. 
  Merwin 
  — 
  Interpolations 
  on 
  Spectrograms. 
  

  

  ' 
  But 
  in 
  some 
  cases, 
  e. 
  g., 
  III. 
  ante, 
  a 
  second 
  approximation 
  is 
  

   required. 
  The 
  resulting 
  formula 
  also 
  is 
  a 
  straight 
  line 
  and 
  is 
  

   used 
  with 
  the 
  same 
  table. 
  The 
  second 
  approximation 
  is 
  based 
  

   upon 
  the 
  considerations 
  which 
  follow. 
  The 
  graph 
  of 
  n' 
  against 
  

   X 
  has 
  gradually 
  changing 
  curvature. 
  We 
  may 
  make 
  use 
  of 
  

   this 
  fact 
  in 
  applying 
  formula 
  8 
  if 
  we 
  simply 
  increase 
  or 
  de- 
  

   crease 
  by 
  the 
  same 
  amount 
  all 
  the 
  values 
  of 
  X 
  along 
  the 
  margin 
  

  

  of 
  the 
  table. 
  Thus 
  in 
  the 
  table 
  n' 
  = 
  — 
  — 
  # 
  01X' 
  2 
  . 
  But 
  

  

  X 
  2 
  - 
  -01 
  

  

  f 
  

   we 
  may 
  write 
  n' 
  = 
  J 
  - 
  — 
  A(X 
  + 
  k) 
  2 
  and 
  reproduce 
  the 
  

  

  values 
  of 
  n' 
  provided 
  k 
  is 
  small 
  compared 
  with 
  X. 
  Trials 
  have 
  

   shown 
  that 
  a 
  small 
  value 
  of 
  h 
  is 
  all 
  that 
  is 
  required 
  to 
  make 
  

   the 
  deviations 
  from 
  formula 
  8 
  very 
  small 
  or 
  negligible 
  ; 
  f* 
  g 
  

   and 
  h 
  do 
  not 
  require 
  evaluation. 
  

  

  Then 
  ^ 
  = 
  R'n' 
  x 
  + 
  h 
  - 
  R' 
  (8b) 
  

  

  Formulas 
  8a 
  and 
  8b 
  apply 
  to 
  a 
  spectrum 
  from 
  a 
  single 
  prism. 
  

   If 
  a 
  train 
  of 
  prisms 
  is 
  used 
  in 
  producing 
  the 
  spectrogram, 
  

   equation 
  3 
  is 
  still 
  a 
  true 
  equation 
  if 
  i 
  is 
  the 
  angle 
  of 
  incidence 
  

   upon 
  the 
  second 
  and 
  each 
  succeeding 
  prism. 
  But 
  i 
  is 
  not 
  con- 
  

   stant, 
  as 
  in 
  the 
  case 
  of 
  a 
  single 
  prism, 
  but 
  it 
  decreases 
  with 
  X. 
  

   This 
  has 
  the 
  effect 
  of 
  slightly 
  changing 
  the 
  relative 
  values 
  of 
  d 
  

   and 
  X. 
  But 
  the 
  changes 
  are 
  taken 
  care 
  of 
  (if 
  necessary) 
  in 
  

   equation 
  8b. 
  

  

  Geophysical 
  Laboratory, 
  

  

  Carnegie 
  Institution 
  of 
  Washington, 
  

  

  Washington, 
  D. 
  C, 
  October 
  9, 
  1916. 
  

  

  