﻿Chemistry 
  and 
  Physics. 
  227 
  

  

  on 
  the 
  whole, 
  so 
  unerringly 
  to 
  the 
  most 
  diverse 
  and 
  nnselected 
  

   material? 
  That 
  it 
  does 
  not 
  apply 
  always 
  and 
  of 
  necessity, 
  may 
  

   be 
  taken 
  as 
  admitted. 
  That 
  it 
  does 
  apply 
  very 
  closely 
  and 
  very 
  

   commonly 
  is 
  a 
  matter 
  of 
  experience. 
  ' 
  ' 
  

  

  If 
  a 
  distribution 
  of 
  errors 
  subject 
  to 
  the 
  regular 
  law 
  — 
  that 
  

   is. 
  occurring 
  proportionately 
  to 
  exp(-/i 
  2 
  x 
  2 
  ) 
  — 
  be 
  taken, 
  if 
  each 
  

   element 
  of 
  this 
  be 
  replaced 
  by 
  another 
  distribution 
  obeying 
  the 
  

   same 
  law. 
  and 
  if 
  the 
  results 
  be 
  then 
  collected 
  in 
  the 
  order 
  of 
  

   their 
  magnitude, 
  a 
  third 
  final 
  distribution 
  will 
  emerge 
  which 
  is 
  

   again 
  subject 
  to 
  the 
  original 
  law. 
  This 
  is 
  the 
  reproductive 
  

   property 
  of 
  the 
  law 
  of 
  error 
  which 
  has 
  been 
  apparently 
  proved 
  

   by 
  a 
  number 
  of 
  writers. 
  By 
  itself 
  such 
  a 
  property 
  leads 
  nowhere, 
  

   for 
  its 
  application 
  is 
  limited 
  to 
  domains 
  already 
  subject 
  to 
  the 
  

   law. 
  It 
  is 
  therefore 
  necessary 
  to 
  make 
  an 
  excursion 
  outside 
  it 
  to 
  

   find 
  the 
  genesis 
  of 
  the 
  law. 
  

  

  Accordingly, 
  let 
  a 
  distribution 
  be 
  taken 
  which 
  does 
  not 
  fol- 
  

   low 
  the 
  law 
  exp(-/i 
  2 
  x 
  2 
  ) 
  strictly 
  but 
  which 
  fluctuates 
  around 
  this 
  

   law; 
  for 
  example, 
  in 
  the 
  manner 
  expressed 
  by 
  exp[(-/i 
  2 
  # 
  2 
  ) 
  

   (1-j-a 
  cos 
  hx)\. 
  If. 
  as 
  above, 
  this 
  be 
  disturbed 
  by 
  a 
  second 
  

   distribution 
  of 
  the 
  same 
  kind, 
  say 
  exp 
  [ 
  (-h 
  2 
  x 
  2 
  ) 
  (1 
  + 
  a' 
  cos 
  k'x) 
  ] 
  , 
  

   a 
  third 
  resultant 
  distribution 
  will 
  be 
  obtained 
  in 
  which 
  the 
  

   fluctuating 
  element 
  tends 
  to 
  efface 
  itself. 
  "Hence 
  if 
  we 
  go 
  on 
  

   piling 
  error 
  upon 
  error, 
  provided 
  each 
  has 
  the 
  fluctuating 
  char- 
  

   acter 
  indicated 
  above, 
  we 
  shall 
  as 
  a 
  limit 
  coirverge 
  to 
  the 
  pure 
  

   law 
  of 
  Gauss." 
  More 
  generally, 
  in 
  order* 
  to 
  obtain 
  an 
  approxi- 
  

   mation 
  to 
  a 
  set 
  of 
  numbers 
  fluctuating 
  about 
  the 
  law 
  of 
  distri- 
  

   bution 
  exp(-/i 
  2 
  ic 
  2 
  ), 
  where 
  h 
  is 
  an 
  adjustable 
  constant, 
  nothing 
  

   more 
  is 
  necessary 
  than 
  to 
  take 
  any 
  holomorphic 
  function 
  as 
  

   originating 
  the 
  error 
  and 
  then 
  to 
  let 
  the 
  frequency 
  curve 
  register 
  

   the 
  number 
  of 
  times 
  individual 
  values 
  occur, 
  disregarding 
  at 
  

   the 
  same 
  time 
  the 
  order 
  in 
  which 
  these 
  values 
  arise 
  naturally. 
  

  

  "If 
  then 
  we 
  suppose 
  that 
  errors 
  are 
  not 
  of 
  mysterious 
  char- 
  

   acter, 
  sui 
  generis, 
  but 
  are 
  simply 
  the 
  mass 
  of 
  numberless 
  neglected 
  

   disturbances, 
  each 
  occurring 
  according 
  to 
  regular 
  law 
  and 
  order 
  

   of 
  its 
  own. 
  it 
  is 
  seen 
  that 
  we 
  obtain 
  the 
  approximation 
  to 
  Gauss's 
  

   law 
  which 
  is 
  necessary 
  to 
  begin 
  with, 
  by 
  the 
  operation 
  of 
  neglect- 
  

   ing 
  the 
  circumstances 
  and 
  order 
  of 
  their 
  origin, 
  and 
  scheduling 
  

   merely 
  in 
  sequence 
  of 
  magnitude 
  the 
  number 
  of 
  times 
  that 
  each 
  

   particular 
  value 
  occurs. 
  It 
  is 
  this 
  operation 
  that 
  is 
  the 
  significant 
  

   act 
  which 
  effaces 
  the 
  individuality 
  of 
  the 
  contributing 
  elements 
  

   and 
  permits 
  us 
  to 
  obtain, 
  apparently 
  from 
  nothing, 
  the 
  law 
  of 
  

   Gauss; 
  for 
  if 
  we 
  go 
  on 
  repeating 
  it 
  for 
  more 
  and 
  more 
  sources 
  

   of 
  error, 
  we 
  obtain 
  the 
  law 
  with 
  greater 
  and 
  greater 
  purity. 
  ' 
  * 
  — 
  

   Phil. 
  Mag., 
  36, 
  347, 
  1918. 
  h. 
  s. 
  u. 
  

  

  8. 
  Atomic 
  Number 
  and 
  Spectral 
  Series. 
  — 
  Ever 
  since 
  the 
  dis- 
  

   covery 
  of 
  series 
  relationships 
  between 
  the 
  lines 
  of 
  ordinary 
  

   spectra 
  (A 
  2000-A 
  8000, 
  say), 
  attempts 
  have 
  been 
  made 
  to 
  find 
  

   an 
  empirical 
  law 
  connecting 
  the 
  constant 
  frequency 
  differences 
  

  

  