﻿426 
  Prof. 
  F. 
  Y. 
  Edgeworth 
  : 
  An 
  Astronomer 
  

  

  stituents 
  of 
  the 
  order 
  of 
  magnitude 
  k 
  4 
  (or 
  & 
  2 
  2 
  ). 
  Whereas 
  the 
  

  

  remainder 
  tends 
  to 
  contain 
  only 
  s 
  constituents 
  of 
  that 
  order 
  ; 
  

  

  s(s 
  — 
  1) 
  

   since 
  the 
  central 
  term 
  in 
  the 
  expansion 
  of 
  k 
  A 
  , 
  viz. 
  6k 
  2 
  k 
  2 
  -^—5 
  — 
  -, 
  

  

  is 
  cancelled 
  by 
  the 
  main 
  term 
  in 
  the 
  expansion 
  of 
  3(s& 
  2 
  ) 
  2 
  *. 
  

   It 
  must 
  be 
  postulated 
  that 
  the 
  mean 
  second 
  and 
  fourth 
  

   powers 
  of 
  deviation 
  are 
  finite 
  : 
  a 
  postulate 
  which 
  may 
  be 
  

   taken 
  for 
  granted 
  when 
  the 
  range 
  of 
  the 
  component 
  

   frequency-functions 
  is 
  finite. 
  The 
  reasoning 
  may 
  be 
  ex- 
  

   tended 
  — 
  if 
  the 
  postulate 
  also 
  is 
  — 
  to 
  further 
  terms 
  in 
  the 
  

   expansion 
  of 
  R. 
  Accordingly 
  Laplace 
  was 
  quite 
  right 
  

   when, 
  referring 
  to 
  symmetrical 
  identical 
  frequency-functions 
  

   with 
  limited 
  range, 
  he 
  affirmed 
  that 
  "taking 
  hyperbolic 
  

   logarithms, 
  we 
  have 
  very 
  approximately 
  (a 
  tres-yeu 
  pres), 
  

   when 
  s 
  is 
  a 
  large 
  number 
  " 
  f 
  for 
  the 
  (Napierian) 
  logarithm 
  

   of 
  R 
  (in 
  the 
  notation 
  above 
  used) 
  an 
  expression 
  equivalent 
  

   to 
  that 
  which 
  has 
  been 
  given 
  above. 
  

  

  The 
  conclusion 
  may 
  be 
  extended 
  to 
  the 
  case 
  of 
  frequency- 
  

   functions 
  not 
  identical 
  ; 
  provided 
  that 
  their 
  mean 
  powers 
  of 
  

   deviation 
  are 
  of 
  the 
  same 
  order. 
  The 
  conditions 
  have 
  been 
  

   stated 
  elsewhere 
  by 
  the 
  present 
  writer 
  with 
  more 
  precision 
  J. 
  

   It 
  is 
  needless 
  to 
  reproduce 
  that 
  statement 
  here, 
  since 
  the 
  con- 
  

   ditions 
  (for 
  even 
  frequency-functions) 
  have 
  been 
  adequately 
  

   stated 
  in 
  a 
  treatise 
  to 
  which 
  Dr. 
  Sampson 
  has 
  referred, 
  

   Poincare's 
  Calcul 
  des 
  Probabilites 
  §. 
  Poincare 
  supposes 
  

   "the 
  functions 
  even, 
  or 
  in 
  other 
  words 
  no 
  systematic 
  

   errors"|| 
  (Calcul, 
  j). 
  184) 
  ; 
  "that 
  the 
  errors 
  are 
  independent" 
  

  

  * 
  For 
  a 
  fuller 
  exposition, 
  see 
  the 
  present 
  writers 
  article 
  on 
  the 
  " 
  Law 
  

   of 
  Error 
  " 
  in 
  the 
  Transactions 
  of 
  the 
  Cambridge 
  Philosophical 
  Society, 
  

   vol. 
  xx. 
  p. 
  42 
  et 
  seq. 
  (1905). 
  

  

  t 
  Theorie 
  analytique 
  des 
  Probabilites, 
  liv. 
  2, 
  Art. 
  18, 
  p. 
  336, 
  National 
  

   edn. 
  1847. 
  

  

  X 
  In 
  the 
  1905 
  Paper 
  referred 
  to 
  in 
  the 
  penultimate 
  note. 
  

  

  § 
  The 
  proof 
  of 
  the 
  law 
  of 
  error 
  by 
  way 
  of 
  approximate 
  identity 
  

   between 
  the 
  mean 
  powers 
  of 
  the 
  representative 
  function 
  and 
  those 
  of 
  

   the 
  actual 
  locus 
  was 
  put 
  forward 
  by 
  the 
  present 
  writer 
  in 
  1905 
  without 
  

   acknowledgment, 
  because 
  without 
  knowledge, 
  of 
  Poincare's 
  similar 
  

   proof 
  for 
  the 
  case 
  of 
  even 
  functions, 
  published 
  in 
  1896. 
  Priority 
  may 
  

   still 
  be 
  claimed 
  for 
  the 
  essay 
  of 
  1905 
  as 
  haying 
  extended 
  that 
  proof 
  to 
  

   odd 
  functions, 
  having 
  proposed 
  several 
  collateral 
  proofs, 
  and 
  having 
  

   carried 
  the 
  approximation 
  beyond 
  the 
  stage 
  at 
  which 
  it 
  was 
  left 
  by 
  

   Poisson 
  (the 
  " 
  second 
  approximation 
  " 
  referred 
  to 
  below 
  note 
  p. 
  428). 
  

  

  || 
  It 
  may 
  be 
  worth 
  while 
  to 
  recall 
  that 
  " 
  symmetrical 
  " 
  and 
  " 
  syste- 
  

   matic 
  " 
  errors 
  are 
  not 
  necessarily 
  coincident. 
  The 
  centre 
  of 
  asymmetrical 
  

   group 
  may 
  not 
  coincide 
  with 
  the 
  true 
  point, 
  and 
  the 
  centroid 
  of 
  an 
  

   unsymmetrical 
  group 
  may. 
  In 
  the 
  case 
  of 
  symmetry 
  as 
  well 
  as 
  of 
  

   asymmetry 
  something 
  must 
  be 
  known 
  or 
  presumed 
  as 
  to 
  the 
  relation 
  

   of 
  the 
  true 
  point 
  to 
  the 
  group 
  as 
  a 
  whole. 
  Cp. 
  Phil. 
  Mag. 
  vol. 
  xvi. 
  

   p. 
  373 
  (1888) 
  and 
  Journal 
  of 
  the 
  Royal 
  Statistical 
  Society, 
  vol. 
  lxxi. 
  

   p. 
  500 
  (1908). 
  

  

  