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  ] 
  

  

  XLIX. 
  An 
  Astronomer 
  on 
  the 
  Law 
  of 
  Error. 
  

   By 
  Professor 
  F. 
  Y. 
  Edgewokth, 
  F.B.A* 
  

  

  rflHE 
  law 
  of 
  error 
  has 
  been 
  disputed 
  by 
  the 
  Astronomer 
  

   JL 
  Royal 
  for 
  Scotland 
  + 
  on 
  two 
  distinct 
  grounds. 
  

  

  I. 
  He 
  finds 
  that 
  the 
  law 
  is 
  not 
  perfectly 
  fulfilled 
  by 
  

   astronomical 
  observations, 
  not 
  even 
  by 
  those 
  made 
  by 
  

   Bradley 
  which 
  Bessel 
  tested, 
  still 
  less 
  by 
  others. 
  This 
  

   verdict 
  does 
  not 
  disconcert 
  the 
  statistician 
  who 
  (after 
  

   Laplace) 
  grounds 
  his 
  expectation 
  of 
  the 
  law 
  on 
  the 
  inter- 
  

   action 
  of 
  numerous 
  independent 
  causes. 
  Where 
  that 
  

   condition 
  is 
  imperfectly 
  fulfilled 
  there 
  is 
  no 
  reason 
  to 
  expect 
  

   the 
  law 
  of 
  error 
  to 
  be 
  realised 
  perfectly, 
  any 
  more 
  than 
  we 
  

   expect 
  a 
  body 
  attracted 
  to 
  another 
  according 
  to 
  the 
  law 
  of 
  

   gravitation 
  to 
  move 
  in 
  a 
  perfect 
  conic 
  when 
  there 
  is 
  a 
  

   resisting 
  medium. 
  In 
  the 
  case 
  of 
  astronomical 
  observations 
  

   we 
  have 
  reason 
  to 
  believe 
  that 
  the 
  conditions 
  will 
  be 
  some- 
  

   times 
  badly, 
  sometimes 
  fairly 
  well 
  fulfilled 
  J. 
  The 
  like 
  is 
  

   true 
  of 
  physical 
  observations 
  generally, 
  with 
  which 
  may 
  be 
  

   classed 
  shots 
  aimed 
  at 
  an 
  object 
  §. 
  Similarly 
  the 
  neigh- 
  

   bouring 
  class 
  of 
  statistics 
  not 
  grouped 
  about 
  an 
  objective 
  

   thing 
  (e. 
  g. 
  statures 
  of 
  a 
  population) 
  fulfil 
  the 
  law 
  more 
  or 
  

   less 
  perfectly. 
  Greater 
  perfection 
  may 
  be 
  expected 
  in 
  those 
  

   classes 
  of 
  phenomena 
  to 
  which 
  Laplace 
  and 
  Poisson 
  confined 
  

   the 
  application 
  of 
  the 
  law, 
  namely 
  magnitudes 
  each 
  of 
  which 
  

   is 
  an 
  average 
  — 
  or 
  more 
  generally 
  a 
  linear 
  function 
  — 
  of 
  

   numerous 
  observations 
  or 
  statistics 
  of 
  the 
  two 
  classes 
  above 
  

   mentioned 
  and 
  (exemplifying 
  such 
  functions 
  ||) 
  occurrences 
  

   at 
  games 
  of 
  chance. 
  The 
  most 
  perfect 
  fulfilment 
  is 
  pre- 
  

   sumably 
  presented 
  in 
  a 
  molecular 
  medley 
  by 
  velocities 
  

   considered 
  as 
  the 
  resultants 
  of 
  innumerable 
  compositions. 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  t 
  " 
  On 
  the 
  Law 
  of 
  Distribution 
  of 
  Errors." 
  By 
  R. 
  A. 
  Sampson. 
  

   Fifth 
  International 
  Congress 
  of 
  Mathematicians, 
  1912, 
  vol. 
  ii. 
  p. 
  163 
  

   et 
  seq. 
  

  

  X 
  Fulfilment 
  of 
  the 
  law 
  is 
  not 
  to 
  be 
  expected 
  where 
  the 
  observations 
  

   are 
  affected 
  by 
  a 
  few 
  dominant 
  sources 
  of 
  error, 
  as 
  pointed 
  out 
  by 
  

   Morgan 
  Crofton 
  in 
  some 
  instructive 
  remarks 
  on 
  the 
  nature 
  of 
  errors 
  

   in 
  astronomical 
  observations/' 
  Philosophical 
  Transactions, 
  1870, 
  p. 
  177. 
  

  

  § 
  In 
  the 
  class 
  of 
  shots 
  may 
  be 
  included 
  guesses 
  e. 
  g. 
  as 
  to 
  the 
  age 
  of 
  

   an 
  individual 
  and 
  even 
  estimates 
  of 
  a 
  less 
  objective 
  magnitude, 
  such 
  as 
  

   the 
  worth 
  of 
  an 
  examination 
  paper 
  ; 
  as 
  to 
  which 
  see 
  Phil. 
  Mag. 
  August 
  

   1890. 
  As 
  to 
  the 
  classification 
  of 
  phenomena 
  obeying 
  the 
  law 
  of 
  error, 
  

   see 
  article 
  on 
  "Probability," 
  Encyclopedia 
  Britannica, 
  11th 
  ed., 
  § 
  117 
  

   et 
  seq. 
  Cp. 
  as 
  to 
  imperfect 
  fulfilment 
  of 
  the 
  conditions, 
  § 
  157. 
  

  

  | 
  E. 
  g. 
  a 
  great 
  number 
  of 
  dice 
  being 
  tossed, 
  the 
  frequency 
  with 
  which 
  

   certain 
  faces 
  turn 
  up. 
  If 
  the 
  dice 
  are 
  not 
  perfectly 
  symmetrical 
  (as 
  

   Weldon 
  found, 
  Phil. 
  Mag. 
  vol. 
  50. 
  1900, 
  p. 
  168), 
  the 
  data 
  have 
  some 
  

   affinity 
  to 
  statistics 
  not 
  representing 
  an 
  objective 
  magnitude. 
  

  

  