﻿on 
  the 
  Law 
  of 
  Error. 
  425 
  

  

  In 
  a 
  4 
  in 
  the 
  separate 
  factors 
  have 
  been 
  omitted 
  " 
  * 
  ; 
  meaning 
  

   no 
  doubt 
  that 
  had 
  the 
  terms 
  been 
  taken 
  into 
  account 
  the 
  

   coefficient 
  o£ 
  a 
  4 
  in 
  R 
  would 
  not 
  have 
  proved 
  approximately 
  

   equal 
  to 
  the 
  corresponding 
  coefficient 
  in. 
  the 
  expansion 
  of 
  e~ 
  i 
  c 
  ". 
  

  

  Now 
  this 
  is 
  exactly 
  what 
  was 
  maintained 
  by 
  the 
  present 
  

   writer 
  in 
  the 
  Philosophical 
  Magazine 
  for 
  1883 
  |. 
  There, 
  and 
  

   elsewhere 
  subsequently*, 
  he 
  has 
  instanced 
  forms 
  of 
  f[x) 
  such 
  

   that 
  when 
  any 
  number 
  of 
  components 
  are 
  superposed 
  in 
  the 
  

   manner 
  of 
  Laplace, 
  the 
  compound 
  does 
  not 
  conform 
  to 
  the 
  

   law 
  of 
  error 
  §. 
  Dr. 
  Sampson, 
  then, 
  is 
  right 
  so 
  far 
  as 
  he 
  

   teaches 
  that 
  the 
  independence 
  |j 
  above 
  postulated 
  is 
  not 
  

   sufficient 
  by 
  itself 
  and 
  without 
  any 
  additional 
  conditions 
  to 
  

   secure 
  the 
  fulfilment 
  of 
  the 
  law 
  of 
  error. 
  But 
  he 
  should 
  

   have 
  added 
  that 
  commonly 
  and 
  practically 
  such 
  additional 
  

   conditions 
  are 
  present. 
  

  

  What 
  those 
  conditions 
  are 
  may 
  be 
  shown 
  by 
  continuing 
  

   the 
  expansion 
  of 
  R. 
  Put 
  k± 
  for 
  the 
  mean 
  fourth 
  power 
  

   of 
  deviations 
  for 
  one 
  of 
  the 
  components 
  from 
  the 
  mean 
  

   (identical 
  with 
  Todhunter's 
  h'" 
  now 
  that 
  the 
  mean 
  is 
  coin- 
  

   cident 
  with 
  the 
  origin). 
  We 
  have 
  then 
  for 
  the 
  logarithm 
  

   of 
  p, 
  

  

  (k 
  2 
  as 
  before 
  denoting 
  the 
  mean 
  square 
  of 
  deviation 
  for 
  any 
  

   •one 
  of 
  the 
  component 
  elements). 
  Whence 
  

  

  io 
  g 
  R= 
  -,Ai* 
  r 
  iVy- 
  1 
  (*,-ay). 
  

  

  In 
  the 
  coefficient 
  of 
  a 
  4 
  in 
  the 
  expansion 
  of 
  R 
  the 
  remainder 
  

   after 
  the 
  first 
  term 
  tends 
  to 
  be 
  negligible 
  in 
  comparison 
  with 
  

   that 
  term 
  as 
  s 
  increases. 
  For 
  the 
  first 
  term 
  contains 
  s 
  2 
  con- 
  

  

  * 
  So 
  the 
  qucBsitum 
  is 
  defined 
  by 
  Todhunter 
  after 
  Poisson. 
  But 
  it 
  

   seems 
  much 
  simpler 
  on 
  the 
  lines 
  of 
  Laplace 
  to 
  investigate 
  the 
  proba- 
  

   bility 
  of 
  an 
  observation 
  occurring- 
  at 
  (or 
  in 
  the 
  immediate 
  neighbourhood 
  

   of) 
  a 
  particular 
  point 
  ; 
  as 
  to 
  which 
  conception 
  see 
  " 
  Law 
  of 
  Error 
  " 
  (by 
  the 
  

   present 
  writer), 
  Camb. 
  Phil. 
  Trans, 
  vol. 
  xx. 
  p. 
  131 
  (1905). 
  Cp. 
  p. 
  40, 
  

   et 
  passim. 
  

  

  t 
  Vol. 
  xvi. 
  pp. 
  304 
  & 
  307. 
  

  

  J 
  « 
  Law 
  of 
  Error," 
  Camb. 
  Phil. 
  Trans, 
  vol. 
  xiv. 
  p. 
  140 
  (1885). 
  

  

  § 
  The 
  functions 
  are 
  of 
  the 
  family 
  specified 
  below 
  (p. 
  429) 
  as 
  " 
  repro- 
  

   ductive.*' 
  They 
  may 
  be 
  expanded 
  in 
  ascending 
  powers 
  of 
  x— 
  by 
  a 
  series 
  

   of 
  integrations 
  with 
  respect 
  to 
  a 
  between 
  limits 
  oo 
  and 
  0. 
  

  

  || 
  Absence 
  of 
  interdependence 
  or 
  correlation, 
  both 
  between 
  the 
  several 
  

   component 
  small 
  errors 
  which 
  make 
  up 
  a 
  total 
  or 
  composite 
  error 
  of 
  

   observation 
  and 
  also 
  between 
  successive 
  composite 
  errors 
  ; 
  these 
  terms 
  

   being 
  used 
  in 
  a 
  wide 
  sense 
  so 
  as 
  to 
  cover 
  the 
  case 
  where 
  the 
  compound 
  

   is 
  an 
  average 
  (or 
  other 
  linear 
  function) 
  and 
  the 
  components 
  are 
  (errors 
  of) 
  

   observations 
  or 
  the 
  same 
  divided 
  by 
  n 
  the 
  number 
  of 
  observations 
  

   {cp. 
  above, 
  p. 
  423 
  note) 
  ; 
  and 
  where 
  the 
  "error" 
  is 
  not 
  a 
  deviation 
  from 
  

   an 
  objective 
  magnitude. 
  

  

  