﻿430 
  An 
  Astronomer 
  on 
  the 
  Law 
  of 
  Error. 
  

  

  Dr. 
  Sampson 
  is 
  thrown 
  off 
  the 
  track 
  by 
  a 
  little 
  slip 
  in 
  a 
  

   mathematical 
  operation 
  such 
  as 
  the 
  best 
  may 
  incur 
  when 
  

   not 
  put 
  on 
  their 
  guard 
  by 
  prior 
  knowledge 
  of 
  the 
  subject. 
  

   Dr. 
  Sampson 
  experiments 
  on 
  frequency-functions 
  of 
  the 
  

  

  form—-- 
  — 
  (or 
  the 
  more 
  general 
  form 
  which 
  is 
  presented 
  

  

  w(l 
  + 
  ar) 
  v 
  ° 
  r 
  

  

  when 
  we 
  take 
  a 
  for 
  the 
  parameter 
  of 
  x 
  *) 
  . 
  Here 
  it 
  may 
  be 
  

   observed 
  0(x) 
  is 
  of 
  the 
  form 
  e~ 
  a 
  (or 
  e~ 
  aa 
  ), 
  and 
  accordingly 
  

   we 
  know 
  a 
  priori 
  that 
  the 
  continued 
  superposition 
  of 
  a 
  com- 
  

   ponents 
  of 
  the 
  type 
  -— 
  ; 
  ^ 
  will 
  have 
  for 
  result 
  -=- 
  — 
  «, 
  • 
  

  

  r 
  Jr 
  tt{1 
  + 
  x 
  2 
  ) 
  a 
  2 
  + 
  x 
  2 
  

  

  Dr. 
  Sampson, 
  employing 
  Morgan 
  Crofton's 
  method 
  of 
  com- 
  

   position 
  by 
  way 
  of 
  integration, 
  finds 
  this 
  result 
  true 
  for 
  two 
  

   components 
  ; 
  but 
  not 
  true 
  in 
  general, 
  for 
  any 
  number 
  of 
  

   components. 
  That 
  is, 
  in 
  our 
  notation, 
  if 
  the 
  frequency- 
  

   function 
  — 
  — 
  =r 
  is 
  compounded 
  with 
  — 
  7 
  - 
  r 
  - 
  t 
  — 
  ~- 
  , 
  the 
  

  

  tt{1 
  + 
  x 
  2 
  ) 
  l 
  nr(a 
  2 
  + 
  x 
  2 
  ) 
  ' 
  

  

  result 
  is 
  

  

  1 
  a 
  + 
  1 
  , 
  1 
  

  

  7r 
  (a 
  + 
  l) 
  z 
  + 
  x 
  z 
  

  

  but 
  not 
  for 
  larger 
  values 
  of 
  a. 
  

  

  In 
  Dr. 
  Sampson's 
  own 
  symbols, 
  <B) 
  the 
  abscissa, 
  and 
  <E> 
  the 
  

   form 
  of 
  the 
  resultant 
  frequency-curve, 
  

  

  dA' 
  1 
  

  

  "^(©HaTr-sf 
  ~ 
  2 
  

  

  + 
  A' 
  2 
  *l+(0-A') 
  

  

  2a7r- 
  1 
  ® 
  2 
  

  

  © 
  4 
  + 
  2© 
  2 
  (a 
  2 
  -rl) 
  + 
  (a 
  2 
  -l) 
  s 
  

  

  (loc. 
  cit. 
  p. 
  170) 
  

  

  This 
  result 
  is 
  evidently 
  untenable 
  ; 
  since 
  it 
  imports 
  that 
  if 
  

   we 
  put 
  together 
  two 
  frequency-curves, 
  each 
  symmetrical 
  

   about 
  a 
  central 
  point 
  and 
  thence 
  descending 
  continuously, 
  

   the 
  central 
  ordinate 
  for 
  the 
  compound 
  will 
  be 
  zero 
  ! 
  But 
  if 
  

   we 
  resolve 
  the 
  expression 
  above 
  equated 
  to 
  <£>(©) 
  into 
  two- 
  

   rational 
  fractions 
  according 
  to 
  the 
  general 
  rule 
  for 
  inte- 
  

   gration, 
  we 
  shall 
  find 
  that 
  while 
  the 
  denominator 
  of 
  the 
  

   result 
  is 
  rightly 
  given 
  by 
  Dr. 
  Sampson, 
  the 
  numerator 
  ought 
  

   to 
  have 
  been 
  (a 
  + 
  l)\_® 
  2 
  + 
  (a 
  2 
  — 
  l) 
  2 
  ]; 
  the 
  expression 
  in 
  square 
  

   brackets 
  being 
  a 
  factor 
  of 
  the 
  denominator. 
  Dividing 
  out 
  

   we 
  obtain, 
  as 
  we 
  ought, 
  for 
  the 
  result 
  

  

  1 
  a 
  + 
  1 
  

   7rS 
  2 
  + 
  (a+l) 
  2 
  ' 
  

  

  * 
  Substituting 
  in 
  the 
  last-written 
  expression 
  for 
  x, 
  x/a 
  and 
  dividing 
  

   the 
  expression 
  by 
  a 
  (substituting 
  for 
  dx, 
  dxja). 
  

  

  