﻿Series 
  Law 
  of 
  Resonance 
  Spectra. 
  255 
  

  

  If 
  this 
  condition 
  held 
  rigorously 
  the 
  series 
  would 
  be 
  

   represented 
  by 
  the 
  formula 
  

  

  1 
  m(m— 
  1) 
  

   ^—=183075-2132?7i 
  + 
  13— 
  S 
  ', 
  

  

  in 
  which 
  \ 
  m 
  is 
  the 
  wave-length 
  of 
  the 
  doublet 
  o£ 
  the 
  mth 
  

   order, 
  2132 
  is 
  the 
  frequency 
  difference 
  between 
  orders 
  and 
  

   + 
  1, 
  13 
  the 
  constant 
  second 
  difference 
  of 
  frequency, 
  and 
  m 
  the 
  

   order 
  of 
  the 
  doublet. 
  The 
  most 
  accurate 
  value 
  or! 
  the 
  second 
  

   constant 
  would 
  be 
  obtained 
  by 
  calculating 
  it 
  from 
  a 
  doublet 
  

   of 
  high 
  order, 
  as 
  a 
  small 
  error 
  would 
  be 
  enormously 
  magnified 
  

  

  by 
  the 
  term 
  — 
  L 
  Z 
  the 
  value 
  of 
  which 
  is 
  351 
  for 
  the 
  

  

  twenty- 
  seventh 
  order. 
  

  

  Calculating 
  the 
  constants 
  2130 
  and 
  12*2 
  from 
  orders 
  and 
  

   5 
  gave 
  calculated 
  values 
  of 
  l/\ 
  which 
  differed 
  from 
  the 
  

   observed 
  by 
  the 
  following 
  amounts 
  : 
  — 
  

  

  et 
  Order. 
  

  

  Difference. 
  

  

  Doublet 
  Order. 
  

  

  Difference 
  

  

  1 
  

  

  

  

  7 
  

  

  + 
  2 
  

  

  3 
  

  

  +1 
  

  

  10 
  

  

  + 
  4 
  

  

  4 
  

  

  -1 
  

  

  13 
  

  

  + 
  14 
  

  

  5 
  

  

  -1 
  

  

  18 
  

  

  + 
  65 
  

  

  5 
  

  

  

  

  23 
  

  

  +166 
  

  

  6 
  

  

  -1 
  

  

  27 
  

  

  +284 
  

  

  The 
  large 
  discrepancies 
  in 
  the 
  higher 
  orders 
  are 
  due 
  to 
  

   incorrect 
  determination 
  of 
  the 
  constants. 
  In 
  spite 
  of 
  this, 
  

   though, 
  the 
  series 
  is 
  well 
  represented 
  up 
  to 
  the 
  seventh 
  

   order. 
  The 
  following 
  formula 
  gave 
  the 
  best 
  results 
  over 
  the 
  

   entire 
  range 
  : 
  — 
  

  

  T- 
  = 
  183075 
  - 
  2131-414 
  = 
  m 
  - 
  12*734— 
  ^ 
  l 
  . 
  

  

  m 
  

  

  The 
  values 
  given 
  in 
  the 
  table 
  were 
  calculated 
  by 
  this 
  for- 
  

   mula 
  and, 
  as 
  will 
  be 
  seen, 
  the 
  agreement 
  is 
  good 
  up 
  to 
  the 
  

   doublet 
  of 
  the 
  fifteenth 
  order. 
  There 
  is 
  a 
  small 
  discrepancy 
  

   in 
  the 
  orders 
  1, 
  3, 
  and 
  4, 
  which 
  appears 
  to 
  be 
  inevitable 
  if 
  

   the 
  constants 
  are 
  so 
  chosen 
  as 
  to 
  make 
  the 
  formula 
  cover 
  a 
  

   wide 
  range. 
  Of 
  course, 
  the 
  formula 
  is 
  not 
  correct 
  for 
  the 
  

   entire 
  series, 
  and 
  though 
  we 
  have 
  tried 
  formulae 
  involving 
  

   higher 
  powers 
  of 
  m 
  than 
  the 
  square, 
  we 
  have 
  been 
  unable 
  to 
  

   develop 
  anything 
  superior 
  to 
  the 
  one 
  given. 
  The 
  discovery 
  

   of 
  the 
  fact 
  that 
  the 
  frequency 
  difference 
  between 
  the 
  com- 
  

   ponents 
  of 
  the 
  doublets 
  is 
  a 
  constant, 
  has 
  been 
  of 
  assistance 
  

  

  T2 
  

  

  