﻿( 
  541 
  ) 
  

  

  XXXIV 
  .—Summation 
  of 
  a 
  Compound 
  Series, 
  and 
  its 
  Application 
  to 
  a 
  Problem 
  in 
  

  

  Probabilities. 
  By 
  Bishop 
  Terrot. 
  

  

  (Read 
  21st 
  February 
  1853.) 
  

  

  The 
  series 
  proposed 
  for 
  solution 
  in 
  the 
  following 
  paper 
  is 
  — 
  

  

  (m 
  — 
  q.m— 
  q 
  — 
  1 
  m 
  — 
  q+p 
  + 
  1) 
  x 
  (1 
  .2. 
  3 
  

  

  + 
  (m 
  — 
  q 
  — 
  1 
  . 
  m— 
  q 
  — 
  2. 
  . 
  . 
  m 
  — 
  q+p) 
  x 
  (2. 
  3.4 
  

  

  q) 
  

  

  q 
  + 
  l) 
  

  

  (A) 
  

  

  + 
  (P-P~ 
  1 
  1 
  x(m— 
  p 
  . 
  m— 
  p 
  + 
  1 
  . 
  . 
  m—p 
  + 
  q 
  + 
  1) 
  

  

  The 
  law 
  of 
  this 
  series 
  is 
  manifest. 
  Each 
  term 
  is 
  the 
  product 
  of 
  two 
  factorials, 
  

   the 
  first 
  consisting 
  of 
  p, 
  and 
  the 
  latter 
  of 
  q 
  factors. 
  And 
  in 
  each 
  successive 
  term, 
  

   the 
  factors 
  of 
  the 
  first 
  factorial 
  are 
  each 
  diminished 
  by 
  one, 
  and 
  those 
  of 
  the 
  latter 
  

   increased 
  by 
  one. 
  

  

  Let 
  there 
  be 
  a 
  series, 
  X„Y 
  1 
  + 
  X 
  ft 
  _ 
  1 
  Y 
  2 
  + 
  X 
  l 
  Y. 
  

  

  where, 
  Y 
  2 
  = 
  Y 
  1 
  + 
  a 
  2 
  , 
  Y 
  3 
  = 
  Y 
  2 
  + 
  a 
  2 
  = 
  Y 
  x 
  + 
  a 
  x 
  + 
  a 
  2 
  , 
  and 
  so 
  on. 
  

  

  Then 
  the 
  series 
  =X„ 
  x 
  Y 
  t 
  

  

  + 
  X„_ 
  x 
  xYj 
  + 
  Aj 
  

  

  + 
  X„_ 
  2 
  x 
  Y 
  l 
  + 
  a 
  x 
  + 
  A 
  2 
  , 
  

   &c. 
  

  

  = 
  2X„xY 
  1 
  + 
  2X„_,x 
  A 
  1 
  + 
  2X 
  n 
  _ 
  2 
  x 
  A 
  2 
  + 
  &C. 
  

  

  where 
  2 
  X„ 
  means 
  the 
  sum 
  of 
  all 
  the 
  terms 
  of 
  X 
  from 
  X 
  x 
  to 
  X„ 
  inclusive. 
  

   Let 
  us 
  then, 
  in 
  the 
  first 
  place, 
  take 
  the 
  differences 
  of 
  the 
  second 
  factorials- 
  

  

  -(1.2.3 
  ?) 
  + 
  (2.3.4 
  ? 
  + 
  l) 
  = 
  (2.3.4. 
  

  

  -(2.3.4. 
  . 
  . 
  .? 
  + 
  l) 
  + 
  (3.4.5 
  ? 
  + 
  2) 
  = 
  (3.4.5. 
  

  

  &c. 
  &c. 
  

  

  Hence 
  the 
  sum 
  of 
  the 
  whole 
  series 
  = 
  

  

  2 
  (m 
  — 
  q 
  .m 
  — 
  q—\ 
  ....... 
  m—p 
  + 
  q 
  + 
  1) 
  .1.2.3 
  

  

  + 
  2(m— 
  q 
  — 
  1 
  . 
  m 
  — 
  q 
  — 
  2 
  m—p 
  + 
  q) 
  . 
  2 
  . 
  3 
  .4 
  . 
  

  

  . 
  . 
  q).q 
  

  

  . 
  . 
  q 
  + 
  l).q 
  

  

  q-l.q 
  

  

  ■ 
  9-Q 
  

  

  + 
  2 
  (m 
  — 
  q 
  — 
  2 
  .m 
  — 
  q 
  — 
  3 
  m— 
  p 
  + 
  q 
  — 
  1). 
  3 
  . 
  4 
  5 
  q+l.q 
  

  

  &c. 
  &c. 
  

  

  VOL. 
  XX. 
  PART 
  IV. 
  7 
  G 
  

  

  (B) 
  

  

  