﻿542 
  BISHOP 
  TERROT 
  ON 
  THE 
  SUMMATION 
  OF 
  A 
  COMPOUND 
  SERIES, 
  

  

  Integrating 
  then 
  each 
  line 
  separately, 
  we 
  have 
  the 
  sum 
  

  

  _ 
  q 
  , 
  \ 
  

  

  ~p 
  + 
  l 
  ' 
  m 
  ~ 
  $ 
  + 
  1 
  - 
  m 
  -q 
  m—p 
  + 
  q 
  + 
  lxl 
  .2.3 
  q 
  — 
  l 
  

  

  + 
  q 
  

  

  p 
  + 
  l' 
  m 
  — 
  q.m 
  — 
  q 
  — 
  1 
  m— 
  p 
  + 
  gx2.3.4 
  q 
  

  

  + 
  q 
  

  

  (C) 
  

  

  p 
  + 
  ~X 
  m 
  — 
  q 
  — 
  ^-m-q 
  — 
  l 
  • 
  ■ 
  . 
  m— 
  p 
  + 
  q 
  — 
  1 
  x3 
  .4 
  . 
  5 
  g 
  + 
  1 
  

  

  &c. 
  &c. 
  

  

  If 
  again 
  we 
  treat 
  this 
  form 
  as 
  we 
  have 
  done 
  the 
  original, 
  by 
  taking 
  the 
  

   (Inferences 
  of 
  the 
  second 
  factorials 
  as 
  they 
  now 
  stand, 
  and 
  again 
  integrating, 
  we 
  

   reproduce 
  the 
  sum 
  in 
  the 
  form 
  

  

  q.q-1 
  

  

  p 
  + 
  l.p 
  + 
  2 
  m 
  ~y 
  + 
  2 
  - 
  m 
  -~q 
  + 
  1 
  • 
  ■ 
  ■ 
  ' 
  m-p 
  + 
  q 
  + 
  lxl.2.3. 
  . 
  . 
  . 
  q-2 
  

  

  + 
  q-q-i 
  , 
  • 
  (D) 
  

  

  f 
  +l.p 
  + 
  2 
  ' 
  m 
  ~ 
  q 
  + 
  1 
  - 
  m 
  ~ 
  q 
  m-p 
  + 
  qx2.3.4:. 
  . 
  . 
  . 
  q-1 
  

  

  &c. 
  &c. 
  

  

  It 
  appears, 
  then, 
  that 
  we 
  may 
  continue 
  this 
  differentiation 
  on 
  the 
  one 
  side 
  

   q 
  times, 
  and 
  integration 
  on 
  the 
  other 
  # 
  + 
  1 
  times; 
  and 
  that 
  at 
  each 
  succeeding 
  

   operation, 
  an 
  additional 
  next 
  lower 
  factor 
  will 
  be 
  introduced 
  into 
  the 
  numerator 
  

   of 
  the 
  fractional 
  coefficient, 
  and 
  an 
  additional 
  next 
  highest 
  into 
  the 
  denominator. 
  

   And 
  after 
  q 
  differentiations, 
  the 
  last 
  factorials 
  will 
  all 
  become 
  unity; 
  and, 
  the 
  

   middle 
  factorial 
  having 
  acquired 
  an 
  additional 
  higher 
  factor 
  at 
  each 
  of 
  q 
  + 
  1 
  inte- 
  

   grations, 
  we 
  have 
  for 
  the 
  sum 
  of 
  the 
  series 
  — 
  

  

  q 
  -q 
  — 
  l 
  . 
  q 
  — 
  2 
  .... 
  1 
  

  

  p 
  + 
  l.p 
  + 
  2 
  . 
  .'.' 
  .'p 
  + 
  q+1 
  xm 
  + 
  l.m 
  m 
  -q+p 
  + 
  l 
  j 
  . 
  . 
  • 
  -(E) 
  

  

  II. 
  

  

  The 
  Problem 
  in 
  Probabilites 
  to 
  which 
  the 
  foregoing 
  summation 
  is 
  applicable, 
  

   is 
  the 
  following 
  : 
  — 
  

  

  Suppose 
  an 
  experiment 
  concerning 
  whose 
  inherent 
  probability 
  of 
  success 
  we 
  

   know 
  nothing, 
  has 
  been 
  made 
  p 
  + 
  q 
  times, 
  and 
  has 
  succeeded 
  p 
  times, 
  and 
  failed 
  

   q 
  times, 
  what 
  is 
  the 
  probability 
  of 
  success 
  on 
  the 
  p 
  + 
  q 
  + 
  V 
  th 
  trial. 
  

  

  This 
  Problem 
  is 
  interesting, 
  because 
  it 
  tends 
  to 
  the 
  discovery 
  of 
  a 
  rational 
  

   measure 
  for 
  those 
  expectations 
  of 
  success 
  which 
  constitute 
  the 
  motive 
  for 
  a 
  large 
  

   portion 
  of 
  human 
  actions. 
  The 
  force 
  of 
  such 
  expectations 
  commonly 
  depends, 
  

   not 
  upon 
  reason, 
  but 
  upon 
  temperament 
  ; 
  and, 
  according, 
  as 
  a 
  man 
  is 
  naturally 
  

   sanguine 
  or 
  the 
  reverse, 
  so 
  in 
  all 
  the 
  contingencies 
  of 
  life, 
  does 
  he 
  over-estimate 
  

   or 
  under-estimate 
  the 
  chances 
  in 
  his 
  favour. 
  

  

  It 
  would 
  be 
  going 
  much 
  too 
  far 
  to 
  think, 
  that 
  we 
  can 
  give 
  an 
  algebraic 
  formula, 
  

   by 
  the 
  application 
  of 
  which 
  a 
  man 
  may, 
  in 
  every 
  practical 
  case, 
  correct 
  his 
  

   natural 
  tendency 
  to 
  error, 
  and 
  arrive 
  at 
  a 
  strictly 
  rational 
  amount 
  of 
  expectation. 
  

  

  

  