﻿424 
  Prof. 
  F. 
  Y. 
  Edgeworth 
  : 
  An 
  Astronomer 
  

  

  the 
  law 
  of 
  frequency 
  is 
  one 
  and 
  the 
  same 
  for 
  all 
  the 
  com- 
  

   ponents, 
  say/(#). 
  Let 
  this 
  function 
  be 
  symmetrical 
  about 
  

   a 
  point 
  which 
  is 
  taken 
  as 
  the 
  origin. 
  Further, 
  let 
  the 
  

   weights, 
  called 
  by 
  Dr. 
  Sampson 
  after 
  Todhunter* 
  y, 
  be 
  equal. 
  

   Then 
  in 
  the 
  place 
  of 
  pi 
  in 
  Dr. 
  Sampson's 
  (and 
  Todhunter's) 
  

   notation 
  we 
  have 
  

  

  i 
  

  

  f{x) 
  COS 
  axda, 
  

  

  where 
  a 
  and 
  —a 
  are 
  the 
  limits 
  which 
  the 
  errors 
  cannot 
  

   exceed 
  f. 
  Then, 
  since 
  

  

  ( 
  a 
  f(x)da 
  

  

  =l-* 
  2 
  p 
  2 
  +..., 
  

  

  if 
  we 
  put 
  k 
  2 
  for 
  the 
  mean 
  square 
  of 
  deviation 
  from 
  the 
  mean,, 
  

   that 
  is 
  twice 
  Dr. 
  Sampson's 
  h 
  2 
  §. 
  If 
  5 
  is 
  the 
  number 
  of 
  the 
  

   components, 
  we 
  have 
  

  

  " 
  Then 
  approximately," 
  as 
  Todhunter 
  has 
  it 
  in 
  his 
  version 
  

   of 
  Poisson's 
  reasoning 
  || 
  , 
  

  

  R 
  = 
  e-y 
  2a 
  ~, 
  where 
  c 
  2 
  = 
  2sk 
  2 
  . 
  

  

  Accordingly 
  the 
  required 
  expression 
  for 
  the 
  frequency 
  of 
  

   observations 
  between 
  assigned 
  values 
  of 
  the 
  abscissa 
  is 
  given 
  

   by 
  multiplying 
  R 
  upon 
  a 
  certain 
  function 
  of 
  a 
  and 
  the 
  said 
  

   values 
  of 
  the 
  abscissa, 
  and 
  integrating 
  the 
  expression 
  thus 
  

   formed 
  with 
  respect 
  to 
  a 
  between 
  limits 
  00 
  and 
  0. 
  To 
  this 
  

   procedure 
  it 
  is 
  objected 
  by 
  Dr. 
  Sampson 
  that 
  "the 
  terms 
  

  

  * 
  See 
  Todhunter, 
  " 
  History 
  of 
  the 
  Theory 
  of 
  Probability," 
  Art. 
  1002. 
  

  

  t 
  In 
  Todhunter's 
  version 
  respectively 
  a 
  and 
  b. 
  

  

  X 
  This 
  condition 
  seems 
  to 
  obviate 
  Dr. 
  Sampson's 
  objection. 
  u 
  If 
  the 
  

   arbitrary 
  distributions 
  fi(x) 
  have 
  any 
  zero 
  — 
  and 
  this 
  is 
  not 
  excluded 
  by 
  

   the 
  process 
  of 
  demonstration 
  — 
  I 
  do 
  not 
  see 
  how 
  they 
  can 
  fail 
  to 
  re- 
  

   appear 
  as 
  zero 
  in 
  the 
  product 
  R 
  " 
  (loc. 
  cit. 
  p. 
  168, 
  par. 
  1). 
  

  

  § 
  Loc. 
  cit. 
  p. 
  167, 
  par. 
  3 
  ; 
  " 
  h 
  " 
  is 
  used 
  differently 
  elsewhere, 
  p. 
  167, 
  

   par. 
  1, 
  pp. 
  169-172. 
  

  

  |l 
  Op. 
  cit. 
  p. 
  565 
  ; 
  Todhunter's 
  Y 
  corresponding 
  to 
  Dr. 
  Sampson's 
  and 
  

   our 
  R, 
  and 
  Todhunter's 
  k 
  2 
  to 
  our 
  ic 
  2 
  and 
  to 
  twice 
  Dr. 
  Sampson's 
  Sfo 
  2 
  

   (when 
  the 
  A's 
  are 
  identical). 
  

  

  