﻿IN 
  THE 
  DIFFERENTIAL 
  CALCULUS. 
  49 
  

  

  / 
  o\ 
  — 
  n 
  n 
  + 
  a 
  + 
  l 
  ( 
  1 
  d 
  \ 
  n 
  n 
  — 
  a 
  — 
  1 
  /a 
  \ 
  

  

  a 
  result 
  which 
  is 
  true, 
  whether 
  n 
  be 
  integral 
  or 
  fractional. 
  

  

  The 
  value 
  of 
  w 
  from 
  Equation 
  (40 
  is 
  easily 
  found, 
  and 
  is 
  given 
  by 
  Mr 
  Boole 
  

   in 
  the 
  form 
  

  

  whence 
  that 
  of 
  u 
  is 
  found. 
  

  

  Let 
  us 
  take, 
  as 
  our 
  next 
  example, 
  the 
  equation 
  which 
  has 
  been 
  discussed 
  by 
  

   M. 
  Poisson 
  in 
  the 
  Journal 
  deVEcole 
  Polytechnique, 
  cah. 
  17, 
  p. 
  614 
  ; 
  and 
  by 
  Profes- 
  

   sor 
  Boole 
  in 
  the 
  Philosophical 
  Transactions 
  for 
  1844, 
  p. 
  254. 
  

  

  Ex.4. 
  

  

  'Q-*)7? 
  + 
  {^-(^-P 
  + 
  I) 
  X 
  }f 
  x 
  -2n(2n-p)u=0. 
  . 
  . 
  . 
  (1.) 
  

  

  The 
  symbolical 
  form 
  of 
  this 
  equation 
  is 
  

  

  M 
  _(D 
  + 
  2,-2)(D 
  + 
  2„-2- 
  y) 
  ?l 
  = 
  

  

  Let 
  u 
  = 
  f(--j) 
  v 
  ■ 
  ■ 
  ■ 
  ■ 
  ( 
  3 
  0i 
  t^n 
  

  

  /(-D.- 
  ^»-gfe»-'-^ 
  /(-°,i)^. 
  = 
  o.. 
  w 
  

  

  This 
  last 
  equation 
  may 
  be 
  reduced 
  to 
  an 
  integrable 
  form 
  in 
  various 
  ways 
  : 
  

  

  i 
  r> 
  t 
  w 
  D 
  tn 
  D 
  +P 
  D+2ra-3 
  / 
  D\ 
  

  

  1. 
  By 
  making 
  /(~ 
  T 
  + 
  1) 
  =pZi 
  • 
  i 
  ) 
  + 
  2n-2-;/ 
  ("2) 
  

  

  or 
  / 
  (~ 
  T 
  ) 
  

  

  2~2 
  /~"2~ 
  w 
  + 
  : 
  

  

  ; 
  ~T 
  + 
  2 
  /"■ 
  2~ 
  w 
  + 
  1 
  + 
  2 
  

  

  /_jD_^_i 
  f__r_p 
  1 
  

  

  M 
  2 
  2 
  2 
  -(2n-p-\)t 
  \ 
  2 
  2 
  2 
  (2n-p^2}i 
  

  

  - 
  e 
  ri 
  or 
  e 
  

  

  / 
  2 
  / 
  2 
  

  

  p+i 
  p-i 
  

  

  / 
  n\P 
  -2' 
  /I 
  ^\ 
  2 
  -2»+2 
  /l 
  rf\ 
  2 
  2n-p-2„ 
  

  

  and 
  equation 
  (4.) 
  becomes 
  

  

  (D 
  + 
  2i»-2)(D 
  + 
  2».-3) 
  g 
  , 
  

   *> 
  D(D-l) 
  * 
  »-«».. 
  (&.; 
  

  

  which 
  is 
  a 
  known 
  form, 
  and 
  thus 
  the 
  given 
  equation 
  is 
  completely 
  solved. 
  

  

  It 
  is 
  evident 
  that 
  our 
  solution 
  reduces 
  the 
  operation 
  to 
  that 
  of 
  ordinary 
  dif- 
  

   ferentiation 
  or 
  integration, 
  when 
  p 
  is 
  an 
  odd 
  integer, 
  positive 
  or 
  negative. 
  By 
  

   varying 
  the 
  process, 
  however, 
  we 
  can 
  obtain 
  other 
  forms 
  of 
  the 
  solution 
  of 
  this 
  

  

  