﻿on 
  the 
  Law 
  of 
  Error. 
  4:21 
  

  

  (op. 
  cit. 
  p. 
  182) 
  ; 
  i; 
  that 
  the 
  individual 
  errors, 
  though 
  

   not 
  following 
  the 
  same 
  law, 
  are 
  practicably 
  (sensiblement) 
  

   of 
  the 
  same 
  order 
  of 
  magnitude, 
  and 
  each 
  contributes 
  little 
  

   to 
  the 
  total 
  error" 
  (op. 
  cit. 
  p. 
  183). 
  • 
  Recapitulating 
  these 
  

   conditions 
  he 
  concludes 
  that 
  " 
  in 
  this 
  case 
  the 
  resultant 
  

   error 
  suivra 
  sensiblement 
  la 
  loi 
  de 
  Gauss 
  " 
  *. 
  

  

  The 
  reasoning 
  whereby 
  the 
  (proper 
  or 
  normal 
  |) 
  law 
  of 
  

   error 
  has 
  been 
  found 
  as 
  an 
  approximation 
  to 
  the 
  actual 
  locus 
  

   (which 
  results 
  from 
  the 
  composition 
  of 
  several 
  independent 
  

   elements) 
  may 
  be 
  extended 
  to 
  obtain 
  a 
  closer 
  approximation 
  

   by 
  taking 
  into 
  account 
  the 
  first 
  of 
  the 
  terms 
  which 
  have 
  

   been 
  neglected. 
  

  

  Putc 
  2 
  =S2£ 
  2 
  , 
  S 
  denoting 
  summation 
  with 
  respect 
  to 
  all 
  

   the 
  elements 
  and 
  k 
  2 
  the 
  mean 
  square 
  of 
  deviation 
  (not 
  now 
  

   identical 
  for 
  all 
  the 
  elements). 
  And 
  for 
  S(& 
  4 
  — 
  3& 
  2 
  2 
  ) 
  put 
  

   K 
  2 
  . 
  Then 
  R 
  may 
  be 
  written 
  

  

  1 
  e-+ 
  c 
  ~ 
  a 
  ' 
  2 
  1 
  1 
  -f 
  j^ 
  a 
  4 
  K 
  2 
  . 
  . 
  . 
  J 
  cos 
  axdx. 
  

  

  The 
  first 
  and 
  main 
  term 
  of 
  the 
  integral 
  is 
  the 
  normal 
  

   error-function 
  

  

  1 
  x 
  2 
  

  

  — 
  7= 
  ex 
  P— 
  ^ 
  sa 
  ) 
  r 
  ^ 
  

  

  The 
  second 
  term, 
  the 
  sought 
  correction, 
  is 
  

  

  f 
  °° 
  1 
  

  

  I 
  e 
  —i 
  c 
  ' 
  a 
  " 
  — 
  f 
  a 
  4 
  K 
  2 
  cos 
  axda-j 
  

  

  Jo 
  ** 
  

  

  which 
  integral 
  may 
  be 
  replaced 
  by 
  — 
  f 
  K 
  2 
  y^. 
  The 
  process 
  

  

  may 
  be 
  extended 
  to 
  terms 
  of 
  smaller 
  orders 
  so 
  as 
  to 
  form 
  a 
  

   descending 
  series, 
  the 
  " 
  generalized 
  law 
  of 
  error 
  " 
  J 
  for 
  even 
  

   functions. 
  

  

  The 
  transition 
  to 
  odd 
  components 
  is 
  effected 
  by 
  obtaining 
  

   likewise 
  an 
  odd 
  descending 
  series; 
  the 
  whole 
  of 
  the 
  odd 
  

   series 
  — 
  like 
  the 
  part 
  of 
  the 
  even 
  series 
  which 
  remains 
  after 
  

   the 
  first 
  main 
  term 
  — 
  tending 
  to 
  vanish 
  as 
  the 
  number 
  of 
  the 
  

  

  * 
  Op. 
  cit. 
  § 
  4, 
  Lecon 
  xvi. 
  Cp. 
  end 
  of 
  § 
  11, 
  Lecon 
  xiv. 
  

  

  t 
  The 
  term 
  " 
  law 
  of 
  error 
  " 
  in 
  this 
  Paper 
  may 
  be 
  understood 
  ac- 
  

   cording 
  to 
  the 
  context 
  either 
  as 
  the 
  " 
  generalized 
  law," 
  or 
  only 
  the 
  

   first 
  and 
  main 
  term 
  of 
  that 
  approximation, 
  the 
  so-called 
  '* 
  Gaussian," 
  or 
  

   " 
  normal," 
  law. 
  

  

  + 
  Camb. 
  Phil. 
  Trans, 
  loc. 
  cit. 
  (1905). 
  Cp. 
  Journal 
  of 
  the 
  Royal 
  

   Statistical 
  Society, 
  1906, 
  " 
  Law 
  of 
  Great 
  Numbers," 
  by 
  the 
  present 
  

   writer. 
  

  

  