﻿46 
  PROFESSOR 
  KELLAND 
  ON 
  A 
  PROCESS 
  

  

  Which 
  equation 
  is 
  reduced 
  to 
  

  

  D(D-l) 
  .... 
  (D-r 
  + 
  l)v 
  = 
  a 
  r 
  e 
  ri 
  v 
  (1.) 
  

  

  by 
  the 
  condition/ 
  (- 
  j 
  +1) 
  = 
  ^ffi 
  / 
  (- 
  7) 
  (2-) 
  

  

  D 
  1 
  

  

  r 
  r 
  

  

  the 
  solution 
  of 
  which 
  equation 
  is 
  

  

  / 
  D_l 
  + 
  

  

  7 
  V 
  ' 
  / 
  / 
  D 
  1 
  . 
  , 
  

  

  /-— 
  +1 
  

  

  / 
  D 
  1 
  

  

  whence 
  - 
  or 
  # 
  r,a 
  V= 
  ; 
  — 
  t^ 
  — 
  i 
  — 
  

  

  _ 
  D_l 
  +1 
  

  

  / 
  r 
  r 
  

  

  / 
  -I-/U-1 
  

  

  / 
  D 
  

  

  /" 
  

  

  = 
  ( 
  — 
  r) 
  x 
  d 
  r 
  x 
  v 
  

  

  Now 
  v 
  is 
  the 
  solution 
  of 
  equation 
  (1) 
  ; 
  or, 
  which 
  is 
  the 
  same 
  thing, 
  of 
  

  

  a 
  V 
  r 
  

  

  = 
  a 
  v 
  

  

  dx 
  r 
  

  

  r=Ac" 
  is 
  a 
  value 
  of 
  v; 
  and 
  hence, 
  finally, 
  

  

  V- 
  ax 
  

  

  =B 
  d 
  r 
  e 
  , 
  where 
  B 
  is 
  an 
  arbitrary 
  constant. 
  

  

  d 
  r 
  v 
  

  

  It 
  is 
  evident 
  that 
  any 
  other 
  of 
  the 
  r 
  roots 
  of 
  the 
  equation 
  — 
  =a 
  v 
  would 
  

  

  dx 
  r 
  

  

  satisfy 
  the 
  conditions, 
  or 
  that 
  any 
  one 
  of 
  the 
  roots 
  of 
  a 
  r 
  will 
  produce 
  the 
  same 
  

   result 
  as 
  a 
  itself. 
  

  

  This 
  example 
  affords 
  an 
  excellent 
  illustration 
  of 
  the 
  process. 
  

  

  Let 
  us 
  take, 
  in 
  the 
  next 
  place, 
  the 
  equation 
  which 
  occurs 
  in 
  the 
  theory 
  of 
  

   the 
  Figure 
  of 
  the 
  Earth 
  ; 
  an 
  equation 
  which 
  has 
  been 
  solved 
  by 
  Professor 
  Boole, 
  

   in 
  his 
  admirable 
  Memoir 
  in 
  the 
  Philosophical 
  Transactions 
  for 
  1844, 
  p. 
  251, 
  but 
  

   whose 
  process 
  is 
  necessarily 
  restricted 
  to 
  integral 
  values 
  of 
  i. 
  The 
  equation 
  is 
  — 
  

  

  T71 
  o 
  d 
  2 
  u 
  , 
  i 
  (i 
  + 
  1) 
  u 
  . 
  

  

  Ex. 
  2. 
  -r-T,—q 
  u~ 
  o 
  =0- 
  

  

  