﻿544 
  BISHOP 
  TERROT 
  ON 
  THE 
  SUMMATION 
  OF 
  A 
  COMPOUND 
  SERIES, 
  

  

  p 
  + 
  l. 
  p 
  + 
  2 
  p 
  + 
  q 
  + 
  1 
  , 
  .. 
  • 
  

  

  = 
  - 
  5-^ 
  - 
  ,ii 
  o 
  q 
  x(m— 
  gf.m-gf-l...m-g-jj) 
  + 
  (lxl.2.3...fl() 
  

  

  m 
  + 
  1 
  .m. 
  . 
  . 
  . 
  m 
  — 
  p 
  — 
  q 
  + 
  1 
  x 
  1 
  . 
  2 
  . 
  3..<? 
  v 
  * 
  * 
  x 
  *v 
  \ 
  y; 
  

  

  But 
  the 
  probability 
  of 
  a 
  white 
  at 
  p 
  + 
  q 
  + 
  l 
  lth 
  drawing 
  on 
  H 
  x 
  is 
  m 
  ~P~ 
  < 
  l 
  

   .-. 
  probability 
  of 
  white 
  derived 
  from 
  H^ 
  is 
  

  

  p 
  + 
  l. 
  p 
  + 
  2 
  .... 
  p 
  + 
  q 
  + 
  1 
  . 
  . 
  \ 
  /1 
  n 
  o 
  n 
  ,^ 
  

  

  *£■ 
  i 
  o 
  q 
  x(m 
  — 
  q.m-q-l...m-q-p)x(l 
  .2.3. 
  ..q) 
  (G) 
  

  

  m 
  + 
  1 
  .m 
  . 
  . 
  . 
  . 
  m 
  — 
  p 
  — 
  q 
  x 
  1 
  .2 
  . 
  6 
  . 
  . 
  . 
  . 
  q 
  v 
  z 
  ■ 
  r/ 
  v 
  *' 
  v 
  * 
  

  

  So 
  probability 
  from 
  H 
  2 
  

  

  «-|_^ 
  , 
  p 
  -J- 
  o 
  _|- 
  ^ 
  

  

  = 
  — 
  -^ 
  — 
  — 
  , 
  o 
  x(m—q 
  — 
  l.m 
  — 
  q-2...m 
  — 
  q—p 
  — 
  l)x 
  (2.3 
  — 
  q+1) 
  

  

  m 
  + 
  1 
  ,m 
  . 
  . 
  . 
  . 
  m—p 
  — 
  q 
  xl 
  . 
  2 
  .6. 
  . 
  . 
  q 
  K 
  2 
  * 
  ' 
  K 
  x 
  ' 
  

  

  And 
  so 
  for 
  all 
  the 
  other 
  hypotheses 
  in 
  succession. 
  

  

  Now 
  this 
  series, 
  omitting 
  for 
  the 
  present 
  the 
  consideration 
  of 
  the 
  fraction 
  

   which 
  is 
  a 
  factor 
  common 
  to 
  them 
  all, 
  is 
  a 
  series 
  of 
  the 
  same 
  form 
  as 
  that 
  summed 
  

   in 
  the 
  last 
  proposition, 
  only 
  that 
  now 
  p 
  + 
  1 
  must 
  be 
  substituted 
  for 
  p. 
  

  

  We 
  have 
  therefore 
  the 
  whole 
  probability 
  of 
  a 
  white 
  at 
  p 
  + 
  q 
  + 
  V 
  th 
  drawing 
  

  

  p 
  + 
  l 
  .p 
  + 
  2 
  . 
  . 
  . 
  p 
  + 
  q 
  + 
  1 
  1 
  .2 
  ... 
  q 
  

  

  m 
  + 
  1 
  . 
  m 
  . 
  . 
  . 
  m—p 
  — 
  q 
  xl 
  . 
  2 
  . 
  . 
  . 
  q 
  p 
  + 
  2 
  . 
  . 
  . 
  p 
  + 
  q 
  + 
  2 
  

  

  7» 
  + 
  L 
  . 
  m 
  . 
  . 
  . 
  m 
  — 
  p 
  — 
  q— 
  — 
  s 
  

  

  r 
  * 
  p+q+2 
  

  

  Note. 
  — 
  It 
  may 
  be 
  worth 
  observing, 
  that, 
  had 
  we 
  summed 
  the 
  original 
  series 
  in 
  

   Prop. 
  1. 
  upwards 
  instead 
  of 
  downwards, 
  we 
  should 
  have 
  got 
  for 
  a 
  first 
  factor 
  

  

  — 
  — 
  t 
  — 
  Vvi 
  — 
  , 
  — 
  tt, 
  which 
  must 
  therefore 
  = 
  —n 
  — 
  '—-^ 
  — 
  ' 
  ' 
  , 
  — 
  n" 
  And 
  

  

  q+1 
  .q 
  + 
  2 
  . 
  . 
  . 
  p 
  + 
  q 
  + 
  V 
  p 
  + 
  l 
  .p 
  + 
  2 
  . 
  . 
  . 
  p 
  + 
  q 
  + 
  1 
  

  

  that 
  these 
  fractions 
  are 
  equal 
  may 
  be 
  proved 
  independently, 
  for 
  if 
  we 
  divide 
  each 
  

   by 
  1.2.3 
  ... 
  p 
  x 
  1.2 
  .3 
  ... 
  q, 
  we 
  have 
  on 
  both 
  sides 
  the 
  same 
  quotient 
  

  

  1 
  

  

  1.2.3 
  .. 
  . 
  p 
  + 
  q 
  + 
  V 
  

  

  There 
  now 
  remains 
  for 
  solution 
  only 
  the 
  first 
  case 
  of 
  the 
  problem 
  in 
  chances, 
  

   that 
  is, 
  to 
  find 
  the 
  probability 
  of 
  drawing 
  a 
  white 
  ball, 
  when 
  m 
  the 
  number 
  of 
  

   balls 
  is 
  given, 
  and 
  p 
  white 
  and 
  q 
  black 
  have 
  already 
  been 
  drawn 
  and 
  returned. 
  

  

  The 
  main 
  object 
  in 
  this 
  case 
  is 
  to 
  sum 
  the 
  series 
  

  

  m- 
  

  

  ^-ll 
  p 
  xr+m-2 
  lp 
  x2' 
  l'.m-V 
  (I) 
  

  

  This 
  may 
  be 
  done 
  much 
  as 
  in 
  the 
  preceding 
  case, 
  by 
  taking 
  the 
  successive 
  

   differences 
  of 
  the 
  right-hand 
  factors 
  till 
  the 
  differences 
  vanish, 
  and 
  multiplying 
  the 
  

   successive 
  terms 
  of 
  the 
  last 
  or 
  q 
  + 
  r 
  th 
  row 
  of 
  differences 
  into 
  the 
  q 
  + 
  l 
  ]th 
  summa- 
  

   tion 
  of 
  the 
  successive 
  terms 
  of 
  the 
  series 
  (l 
  + 
  2 
  p 
  . 
  . 
  . 
  + 
  m-V 
  p 
  ) 
  + 
  (l 
  + 
  2 
  p 
  . 
  . 
  . 
  +m^2^\ 
  

   &c. 
  

  

  This 
  may 
  be 
  sufficiently 
  explained 
  by 
  going 
  through 
  the 
  operation 
  in 
  a 
  low 
  

   particular 
  case. 
  Let 
  p=2, 
  q=3. 
  

  

  