﻿An 
  Astronomer 
  7 
  on 
  the 
  Law 
  of 
  Error. 
  423 
  

  

  II. 
  The 
  ground 
  for 
  expecting 
  that 
  the 
  law 
  o£ 
  error 
  will 
  be 
  

   fulfilled 
  more 
  or 
  less 
  approximately 
  in 
  the 
  preceding 
  cases 
  

   would 
  be 
  cut 
  away 
  if 
  Dr. 
  Sampson's 
  objection 
  to 
  the 
  proof 
  

   of 
  the 
  law 
  were 
  held 
  valid. 
  His 
  attack 
  (loc. 
  cit. 
  p. 
  167) 
  on 
  

   the 
  proof 
  given 
  by 
  Poisson 
  after 
  Laplace 
  strikes 
  at 
  all 
  the 
  

   applications 
  of 
  the 
  law 
  * 
  ; 
  it 
  cannot 
  be 
  limited 
  to 
  the 
  par- 
  

   ticular 
  class 
  of 
  astronomical 
  (or 
  more 
  generally 
  physical) 
  

   observations 
  — 
  to 
  which 
  indeed 
  Laplace 
  f 
  and 
  Poisson 
  did 
  

   not 
  propose 
  to 
  apply 
  the 
  law. 
  The 
  objection 
  here 
  combated 
  

   is 
  not 
  based 
  on 
  the 
  want 
  of 
  that 
  independence 
  which 
  the 
  law 
  

   postulates 
  ; 
  whether 
  as 
  between 
  the 
  (total) 
  errors 
  of 
  succes- 
  

   sive 
  observations, 
  considered 
  as 
  not 
  " 
  accidental 
  " 
  (Sampson, 
  

   loc. 
  cit. 
  p. 
  168), 
  or 
  between 
  the 
  "small 
  errors" 
  (p. 
  166) 
  the 
  

   components 
  J 
  of 
  the 
  entities 
  which 
  may 
  be 
  expected 
  to 
  fulfil 
  

   the 
  law 
  §. 
  It 
  is 
  here 
  conceded 
  that 
  so 
  far 
  as 
  such 
  inde- 
  

   pendence 
  is 
  not 
  present 
  perfect 
  fulfilment 
  of 
  the 
  law 
  is 
  not 
  

   to 
  be 
  expected 
  ; 
  the 
  case 
  will 
  present 
  imperfection 
  of 
  the 
  

   kind 
  admitted 
  under 
  head 
  I. 
  But 
  Dr. 
  Sampson 
  in 
  his 
  

   attack 
  on 
  the 
  theory 
  of 
  Laplace 
  and 
  Poisson 
  [loc. 
  cit. 
  p. 
  167) 
  

   does 
  not 
  dispute 
  the 
  initial 
  stages 
  of 
  the 
  proof 
  in 
  which 
  

   this 
  independence 
  is 
  implied. 
  It 
  is 
  implied 
  that 
  if 
  one 
  

   component 
  error 
  is 
  distributed 
  according 
  to 
  the 
  frequency- 
  

   function 
  fi{x) 
  || 
  and 
  another 
  according 
  to 
  f 
  2 
  {.r), 
  the 
  proba- 
  

   bility 
  that 
  the 
  respective 
  component 
  errors 
  x-i 
  and 
  x 
  2 
  should 
  

   concur 
  is 
  proportionate 
  to 
  /i(#i) 
  xf 
  2 
  (x 
  2 
  ). 
  What 
  the 
  point 
  of 
  

   Dr. 
  Sampson's 
  objection 
  is 
  may 
  be 
  shown 
  by 
  a 
  free 
  version 
  

   of 
  his 
  argument 
  in 
  a 
  simplified 
  case. 
  Let 
  us 
  suppose 
  that 
  

  

  * 
  Including' 
  the 
  method 
  of 
  sampling 
  which 
  is 
  becoming 
  so 
  important 
  

   in 
  social 
  statistics: 
  as 
  to 
  the 
  theory 
  of 
  which 
  see 
  Bowle} 
  T 
  's 
  Presidential 
  

   Address 
  to 
  Section 
  F 
  of 
  the 
  British 
  Association, 
  1906, 
  and 
  as 
  to 
  the 
  

   practice 
  ; 
  Livelihood 
  and 
  Poverty/ 
  1915, 
  by 
  Bowley 
  and 
  another. 
  

  

  t 
  Glaisher 
  more 
  than 
  once 
  remarks 
  on 
  the 
  fact 
  that 
  Laplace 
  did 
  not 
  

   employ 
  his 
  theorem 
  to 
  establish 
  a 
  presumption 
  that 
  observations 
  them- 
  

   selves 
  — 
  as 
  distinguished 
  from 
  averages 
  thereof 
  — 
  fulfil 
  the 
  law 
  of 
  error. 
  

   See 
  ' 
  Memoirs 
  of 
  the 
  Astronomical 
  Society,' 
  vol. 
  xxxix. 
  pp. 
  104, 
  10b' 
  ; 
  

   and 
  ' 
  Monthlv 
  Notices 
  of 
  the 
  Astronomical 
  Society,' 
  vol. 
  xxxiii. 
  p. 
  397, 
  

   par. 
  3 
  (1873). 
  

  

  X 
  \\ 
  nere 
  the 
  entities 
  are 
  averages 
  their 
  components 
  consist 
  of 
  the 
  

   original 
  data, 
  observations 
  or 
  statistics 
  of 
  the 
  kinds 
  described 
  above 
  

   (divided 
  by 
  the 
  number 
  thereof). 
  

  

  § 
  For 
  an 
  example 
  of 
  the 
  first 
  sort 
  of 
  interdependence, 
  see 
  Article 
  on 
  

   " 
  Probability 
  " 
  {Enc. 
  Brit), 
  §157. 
  Where 
  the 
  figures 
  grouped 
  were 
  

   each 
  an 
  average 
  of 
  a 
  set 
  of 
  consecutive 
  observations 
  of 
  the 
  kind 
  instanced, 
  

   the 
  materials 
  would 
  illustrate 
  interdependence 
  of 
  the 
  second 
  kind. 
  

  

  || 
  Meaning 
  that 
  the 
  number 
  of 
  observations 
  which 
  occur 
  between 
  

   x 
  and 
  x+Ax=~Sf 
  l 
  (x)Ax, 
  where 
  N 
  is 
  the 
  total 
  number 
  of 
  observations: 
  

   2f 
  1 
  (x)Ax 
  = 
  l; 
  Ax 
  is 
  small, 
  so 
  that 
  for 
  the 
  purpose 
  in 
  hand 
  it 
  may 
  be 
  

   replaced 
  bv 
  dx. 
  

  

  2G2 
  

  

  