﻿AND 
  ITS 
  APPLICATION 
  TO 
  A 
  PROBLEM 
  IN 
  PROBABILITIES. 
  545 
  

  

  Then 
  the 
  series 
  written 
  perpendicularly 
  is 
  

  

  m-l' 
  2 
  xl 
  Sjm-r'xl 
  2 
  2 
  m-V 
  xl 
  2 
  8 
  m-ll-xl 
  2 
  4 
  m-l'- 
  x 
  1 
  

  

  ^^2i 
  2 
  x8 
  2 
  1 
  1^2 
  |2 
  x7 
  22^2i 
  2 
  x6 
  2 
  3 
  m-2 
  |2 
  x5 
  2j*^2i 
  2 
  x 
  4 
  

  

  ^^T3i 
  2 
  x 
  27 
  = 
  2 
  1 
  ~m^3 
  |2 
  xl9 
  = 
  22^3' 
  2 
  x 
  12 
  = 
  2 
  3 
  m-3 
  |2 
  x6 
  = 
  2 
  i 
  »^3 
  2 
  x 
  1 
  

   ^3T 
  |2 
  x 
  64 
  2 
  1 
  m-4 
  |2 
  x 
  37 
  2 
  2 
  m-4 
  |2 
  x 
  18 
  2 
  3 
  m-4 
  |2 
  x 
  6 
  

   ^5i 
  2 
  x 
  125 
  2jm-5 
  |2 
  x 
  61 
  2 
  2 
  m-5i 
  2 
  x 
  20 
  2 
  3 
  w-5 
  |2 
  x 
  6 
  

  

  &c. 
  &c. 
  &c. 
  &c. 
  

  

  The 
  value 
  of 
  the 
  different 
  sigraas 
  is 
  easily 
  found 
  by 
  the 
  method 
  of 
  finite 
  dif- 
  

   ferences. 
  

  

  Generally, 
  since 
  the 
  differences 
  of 
  l 
  q 
  , 
  2 
  q 
  , 
  3 
  ? 
  , 
  kc, 
  always 
  vanish 
  in 
  the 
  q 
  + 
  l 
  m 
  

   line 
  and 
  after 
  the 
  q 
  ih 
  term 
  of 
  it, 
  the 
  general 
  expression 
  is 
  

  

  2 
  g+1 
  m-V 
  p 
  + 
  d 
  2 
  2 
  g+1 
  m-2l 
  p 
  . 
  . 
  . 
  . 
  d 
  q 
  2 
  q+1 
  m-<f 
  ; 
  

  

  d 
  v 
  d 
  2 
  , 
  d 
  z 
  , 
  &c, 
  signifying 
  the 
  1st, 
  2d, 
  3d, 
  &c, 
  terms 
  of 
  the 
  q 
  + 
  V 
  th 
  row 
  of 
  differences. 
  

   This 
  summation 
  may 
  be 
  applied 
  to 
  find 
  the 
  probability 
  in 
  the 
  case 
  now 
  under 
  

   consideration, 
  for 
  it 
  expresses 
  the 
  a 
  + 
  (3 
  + 
  7, 
  &c, 
  of 
  the 
  preceding 
  case. 
  Applying 
  it 
  

   as 
  we 
  did 
  the 
  value 
  of 
  « 
  + 
  £ 
  + 
  7, 
  &c, 
  there 
  found, 
  we 
  shall 
  find 
  the 
  probability 
  of 
  

   a 
  white 
  ball 
  at 
  the 
  p 
  + 
  q 
  + 
  l 
  lth 
  trial 
  to 
  be 
  

  

  2 
  q+lW 
  ^T 
  p 
  ^ 
  + 
  ^ 
  2 
  2^ 
  1 
  "^^2 
  lf>+1 
  . 
  . 
  . 
  . 
  d 
  q 
  \ 
  + 
  Jn^} 
  p 
  + 
  l 
  

  

  m 
  (\ 
  +1 
  m-V 
  p 
  + 
  d 
  2 
  I 
  q+1 
  m-2^ 
  . 
  . 
  . 
  . 
  d 
  q 
  \^m-q^ 
  ^ 
  

  

  If 
  in 
  be 
  infinite, 
  the 
  expression 
  becomes 
  

  

  (l 
  + 
  d 
  2 
  + 
  .... 
  d 
  q 
  ).2 
  q 
  + 
  x 
  mP+ 
  y 
  _ 
  S^mP* 
  l 
  

   m(l 
  + 
  d 
  2 
  . 
  . 
  . 
  . 
  d 
  q 
  ) 
  .\ 
  +l 
  m 
  p 
  m'2 
  q+ 
  im 
  r> 
  

  

  Bat 
  if 
  x 
  be 
  a 
  quantity 
  varying 
  between 
  the 
  limits 
  0, 
  x, 
  

   And 
  by 
  continuation 
  

  

  l 
  q 
  + 
  x 
  m?+ 
  l 
  = 
  p+\.p 
  + 
  2 
  .... 
  p 
  + 
  q 
  + 
  l 
  _ 
  p 
  + 
  1 
  (L) 
  

  

  mVirf 
  p 
  + 
  2.p 
  + 
  3 
  . 
  . 
  . 
  . 
  p 
  + 
  q 
  + 
  2 
  p+p 
  + 
  2 
  ' 
  ' 
  . 
  

  

  We 
  have 
  thus 
  found 
  the 
  probability 
  in 
  every 
  case 
  of 
  the 
  problem 
  ; 
  the 
  2d 
  and 
  

   4th 
  at 
  H, 
  for 
  the 
  result, 
  being 
  independent 
  of 
  m, 
  must 
  be 
  true 
  for 
  an 
  infinite 
  as 
  

   well 
  as 
  for 
  a 
  finite 
  number. 
  The 
  1st 
  case 
  is 
  solved 
  at 
  K, 
  and 
  the 
  3d 
  at 
  L. 
  

  

  VOL. 
  XX. 
  PART 
  IV. 
  ^ 
  H 
  

  

  