﻿on 
  the 
  Law 
  of 
  Error. 
  429 
  

  

  frequency-curves 
  obtained 
  by 
  grafting 
  an 
  oscillating 
  element 
  

   on 
  the 
  normal 
  law, 
  as 
  thus 
  

  

  6(x) 
  = 
  — 
  = 
  -—~e- 
  h2 
  * 
  2 
  (l 
  '+ 
  a 
  cos 
  hx) 
  *. 
  

  

  rV 
  ' 
  Vtt(1 
  + 
  A) 
  j 
  

  

  He 
  shows 
  that 
  when 
  curves 
  of 
  this 
  type 
  are 
  compounded 
  the 
  

   divergence 
  from 
  the 
  normal 
  law 
  tends 
  to 
  disappear. 
  This 
  

   curiosum 
  may 
  have 
  some 
  bearing 
  on 
  astronomical 
  obser- 
  

   vations. 
  But 
  it 
  may 
  be 
  doubted 
  whether 
  observations 
  

   extending 
  to 
  infinity 
  — 
  other 
  than 
  the 
  law 
  of 
  error 
  — 
  have 
  

   much 
  concrete 
  significance. 
  

  

  The 
  result 
  of 
  continued 
  composition 
  may 
  often 
  be 
  more 
  

   advantageously 
  contemplated 
  by 
  means 
  of 
  Laplace's 
  method 
  

   above 
  indicated. 
  Let 
  the 
  frequency- 
  function 
  tor 
  one 
  element 
  

   be 
  </>i(#), 
  for 
  another 
  §i(x), 
  ranging 
  between 
  given 
  limits, 
  

   which 
  may 
  be 
  infinite. 
  If 
  the 
  functions 
  are 
  even 
  f, 
  

   put 
  p 
  1 
  = 
  (£] 
  (ax) 
  cos 
  ax. 
  And 
  let 
  i 
  p 
  } 
  dx 
  (between 
  extreme 
  

   limits) 
  =0j( 
  «). 
  Let 
  # 
  2 
  ( 
  a 
  ) 
  be 
  formed 
  likewise 
  from 
  </> 
  2 
  (#). 
  

   Then 
  for 
  the 
  compound 
  of 
  the 
  two 
  frequencies 
  we 
  have 
  

  

  j 
  

  

  Jo 
  

  

  #i(a) 
  X 
  6 
  2 
  (a) 
  cos 
  uxda. 
  

  

  This 
  method 
  may 
  be 
  employed 
  to 
  obtain 
  an 
  answer 
  to 
  a 
  

   question 
  which 
  Dr. 
  Sampson 
  has 
  raised, 
  but 
  has 
  not 
  rightly 
  

   answered: 
  namely, 
  what 
  frequ6ncy-curves 
  enjoy 
  the 
  property 
  

   that 
  when 
  two 
  of 
  a 
  family 
  are 
  compounded 
  the 
  result 
  belongs 
  

   to 
  the 
  same 
  familv. 
  Lr. 
  Sampson 
  appears 
  to 
  think 
  that 
  the 
  

   normal 
  law 
  of 
  error 
  is 
  the 
  only 
  curve 
  which 
  possesses 
  this 
  

   property 
  in 
  perfection, 
  " 
  with 
  any 
  generality 
  " 
  (loc. 
  cit. 
  

   p. 
  170). 
  He 
  is 
  not 
  aware 
  that 
  the 
  property 
  appertains 
  to 
  

   a 
  wide 
  class 
  of 
  which 
  the 
  normal 
  law 
  is 
  a 
  species: 
  namely, 
  

   the 
  class 
  for 
  which 
  6(a) 
  (as 
  above 
  defined) 
  is 
  of 
  the 
  form 
  

   exp— 
  a*. 
  There 
  is 
  here 
  disclosed 
  a 
  variant 
  proof 
  of 
  the 
  law 
  

   of 
  error. 
  If 
  there 
  is 
  a 
  final 
  form 
  resulting 
  from 
  continued 
  

   composition, 
  it 
  must 
  be 
  reproductive 
  ; 
  and 
  reproduction 
  is 
  the 
  

   jpropriura 
  of 
  curves 
  for 
  which 
  6(a) 
  is 
  of 
  the 
  form 
  exp— 
  of. 
  

   A 
  further 
  condition 
  (borrowed 
  from 
  Morgan 
  Crof 
  ton) 
  which 
  

   must 
  be 
  fulfilled 
  by 
  the 
  sought 
  form 
  limits 
  t 
  to 
  the 
  value 
  2. 
  

   Using 
  the 
  value 
  of 
  6(a) 
  thus 
  obtained, 
  viz. 
  exp 
  — 
  a 
  2 
  , 
  we 
  

   obtain 
  the 
  normal 
  law 
  %. 
  

  

  * 
  A 
  simple 
  case 
  of 
  the 
  more 
  general 
  type 
  proposed 
  by 
  Dr. 
  Sampson 
  

   (loc. 
  cit. 
  p. 
  170). 
  Note 
  that 
  a 
  is 
  leas 
  than 
  unity, 
  and 
  A 
  is 
  taken 
  so 
  that 
  

   the 
  integral 
  of 
  the 
  function 
  between 
  limits 
  + 
  00 
  and 
  — 
  00 
  is 
  unity. 
  

  

  t 
  As 
  in 
  the 
  case 
  above 
  instanced. 
  For 
  the 
  case 
  of 
  odd 
  functions 
  

   see 
  Camb. 
  Phil. 
  Trans, 
  loc. 
  cit. 
  p. 
  53 
  (1905). 
  

  

  X 
  See 
  Camb. 
  Phil. 
  Trans. 
  1905, 
  " 
  Law 
  of 
  Error," 
  Part 
  1. 
  § 
  4, 
  and 
  

   Appendix, 
  § 
  6. 
  

  

  