﻿428 
  Prof. 
  F. 
  Y. 
  Edgeworth 
  : 
  An 
  Astronomer 
  

  

  components 
  is 
  increased. 
  Thus 
  the 
  first 
  term 
  of 
  the 
  odd 
  

   series 
  may 
  be 
  written 
  

  

  where 
  K 
  l5 
  = 
  S& 
  3 
  *, 
  is 
  the 
  mean 
  third 
  power 
  of 
  deviation 
  for 
  any 
  

   element 
  from 
  its 
  mean 
  value 
  ; 
  the 
  mean 
  value 
  being 
  taken 
  as 
  

   the 
  origin 
  for 
  each 
  of 
  the 
  elements, 
  (and 
  for 
  their 
  sum) 
  f- 
  

   When 
  the 
  differentiation 
  is 
  performed 
  it 
  is 
  seen 
  that 
  the 
  

   term 
  is 
  affected 
  with 
  a 
  coefficient 
  which 
  diminishes 
  as 
  the 
  

   number 
  of 
  components, 
  say 
  s, 
  is 
  increased. 
  For 
  example, 
  

   if 
  each 
  element 
  is 
  a 
  binomial 
  assuming 
  the 
  value 
  or 
  a 
  with 
  

   respective 
  probabilities 
  q 
  and 
  p 
  y 
  

  

  that 
  is 
  of 
  the 
  order 
  1/^/spq. 
  Thus 
  not 
  only 
  Laplace's 
  proof 
  

   of 
  the 
  law 
  of 
  error 
  for 
  even 
  frequency-functions, 
  but 
  also 
  

   Poisson's 
  proof 
  thereof 
  for 
  odd 
  functions, 
  together 
  with 
  his 
  

   determination 
  of 
  a 
  second 
  approximation 
  J, 
  are 
  found 
  to 
  hold 
  

   good 
  upon 
  certain 
  conditions. 
  One 
  who 
  considers 
  that 
  those 
  

   conditions 
  are 
  very 
  generally 
  realized 
  will 
  regard 
  as 
  mis- 
  

   leading 
  the 
  Astronomer 
  Royal's 
  statement 
  : 
  " 
  The 
  conclusion 
  

   is 
  therefore 
  unwarranted 
  and 
  there 
  is 
  no 
  proof 
  at 
  all 
  that 
  

   peculiarities 
  of 
  the 
  functions 
  /efface 
  themselves 
  in 
  the 
  final 
  

   result" 
  (loc. 
  cit. 
  pp. 
  167-8). 
  

  

  III. 
  It 
  remains 
  to 
  notice 
  two 
  other 
  proofs 
  § 
  of 
  Laplace's 
  

   law 
  || 
  which 
  have 
  been 
  likewise 
  approached 
  and 
  missed 
  by 
  

   Dr. 
  Sampson. 
  

  

  There 
  is 
  first 
  Morgan 
  Crofton's 
  proof 
  — 
  either 
  by 
  way 
  of 
  a 
  

   partial 
  differential 
  equation 
  T[, 
  or 
  more 
  directly** 
  — 
  that 
  the 
  

   continued 
  superimposition 
  of 
  frequency-curves 
  of 
  any 
  form 
  

   will 
  result 
  in 
  the 
  normal 
  law 
  of 
  error. 
  Dr. 
  Sampson 
  

   applies 
  Morgan 
  Crofton's 
  method 
  of 
  composition 
  to 
  certain 
  

  

  * 
  It 
  will 
  be 
  noticed 
  that 
  the 
  subscripts 
  of 
  the 
  &'s 
  correspond 
  to 
  

   powers 
  (of 
  the 
  component 
  errors). 
  The 
  subscripts 
  of 
  the 
  K's 
  correspond 
  

   to 
  corrections 
  of 
  the 
  normal 
  function. 
  

  

  t 
  Mutatis 
  mutandis, 
  if 
  the 
  elements 
  are 
  weighted. 
  

  

  % 
  Given 
  by 
  Todhunter, 
  op. 
  cit. 
  pp. 
  567-8. 
  

  

  § 
  Two 
  among 
  several 
  variant 
  proof 
  s 
  which 
  are 
  referred 
  to 
  in 
  the 
  

   article 
  on 
  " 
  Probability" 
  (Ency. 
  Brit.) 
  §§ 
  104-111. 
  

  

  || 
  Laplace 
  rather 
  than 
  Gauss 
  deserves 
  to 
  be 
  the 
  eponym 
  of 
  the 
  law 
  of 
  

   error 
  when, 
  as 
  throughout 
  in 
  this 
  paper, 
  it 
  is 
  considered 
  as 
  resulting 
  

   from 
  the 
  random 
  combination 
  of 
  numerous 
  independently 
  fluctuating 
  

   constituents. 
  

  

  % 
  Cited 
  in 
  the 
  article 
  on 
  "Probability" 
  {Ency. 
  Brit.), 
  § 
  109. 
  

   ** 
  See 
  Phil. 
  Trans. 
  1870. 
  

  

  