﻿48 
  PROFESSOR 
  KELLAND 
  ON 
  A 
  PROCESS 
  

  

  in 
  which 
  form 
  the 
  solution 
  has 
  been 
  given 
  by 
  Professor 
  Boole, 
  when 
  i 
  is 
  a 
  whole 
  

   number. 
  

  

  Ex. 
  3. 
  Solution 
  of 
  the 
  .Equation 
  for 
  Laplace's 
  Functions. 
  

  

  This 
  equation 
  has 
  been 
  reduced 
  by 
  Professor 
  Boole, 
  in 
  the 
  Cambridge 
  Ma- 
  

   thematical 
  Journal, 
  New 
  Series, 
  vol. 
  i., 
  p. 
  15, 
  to 
  the 
  form 
  

  

  ( 
  l 
  -^Jj? 
  + 
  2(«-l) 
  /x^-f 
  {«(» 
  + 
  !)-«(«- 
  1)}»=0 
  ... 
  (1) 
  

   where 
  u, 
  the 
  quantity 
  sought, 
  is 
  connected 
  with 
  v 
  by 
  the 
  equation 
  

  

  a 
  

  

  u 
  = 
  (l-^)~ 
  2 
  v 
  (2). 
  

   Putting 
  fx=e 
  and 
  writing 
  D 
  for 
  ^, 
  we 
  have 
  

  

  {( 
  e 
  - 
  2 
  '_l)D(D-l) 
  + 
  2(a-l)D 
  + 
  »(w 
  + 
  l)-a(a-1)}»=0 
  

  

  or 
  e~ 
  26 
  D 
  (D-l) 
  v 
  -(D-'a 
  + 
  n) 
  (D-a-n-1) 
  v 
  = 
  

  

  or 
  D(D-l)»-(D-a-»-2) 
  (T>-a 
  + 
  n-l)e 
  2t! 
  v 
  = 
  

  

  Let, 
  therefore, 
  v—f 
  ( 
  — 
  k) 
  n> 
  (3), 
  and 
  this 
  equation 
  becomes 
  

  

  D(D-l)/(--^)w-(D-rt-n-2)(D-a 
  + 
  «-l)/(--^ 
  + 
  l) 
  e 
  2 
  ^ 
  W 
  = 
  

  

  D(D-l)w-(D-a-n-l)(D-a-n-2)e 
  2 
  *?v=0 
  (4), 
  provided 
  

   • 
  D 
  1A 
  J)- 
  a 
  -n-l 
  / 
  D\ 
  

  

  D 
  a 
  n 
  1 
  

  

  or 
  

  

  " 
  2 
  " 
  r 
  2 
  n 
  "2 
  ^2 
  /__D\ 
  

   _ 
  D 
  a 
  n 
  ]V 
  \ 
  2) 
  

  

  2^2 
  2 
  2 
  

  

  an 
  equation 
  of 
  which 
  the 
  solution 
  is 
  evidently 
  

  

  /__D 
  a 
  n 
  1 
  

  

  /(-D4t?S 
  (5 
  ° 
  

  

  / 
  2 
  + 
  2 
  2 
  + 
  2 
  

   and 
  by 
  (3) 
  »=/( 
  — 
  gj^ 
  

  

  /_JD 
  « 
  n 
  1 
  

  

  _/ 
  T 
  + 
  2 
  + 
  2 
  + 
  2 
  

  

  /__D 
  «_» 
  1 
  

  

  / 
  T 
  + 
  2~2 
  + 
  2 
  

  

  — 
  «+ 
  w 
  

  

  = 
  (n-a-\)6 
  I 
  A 
  (n-a-l)f 
  ' 
  

  

  / 
  D 
  

  

  /"T 
  

  

  