﻿IN 
  THE 
  DIFFERENTIAL 
  CALCULUS. 
  51 
  

  

  and 
  «=(-2) 
  >* 
  *(-^J 
  

  

  where 
  « 
  is 
  found 
  from 
  the 
  solution 
  of 
  the 
  equation 
  of 
  the 
  first 
  order, 
  

  

  D 
  + 
  2»-2 
  2* 
  ./ 
  D\-i 
  A 
  

  

  This 
  solution 
  is 
  of 
  the 
  ordinary 
  form 
  when 
  n-p 
  is 
  an 
  integer. 
  

  

  This 
  last 
  form 
  appears 
  to 
  have 
  been 
  overlooked 
  by 
  Professor 
  Boole 
  

  

  — 
  -o- 
  ) 
  ° 
  m 
  the 
  first 
  solution. 
  

  

  The 
  reason 
  is, 
  that 
  equation 
  (5) 
  being 
  of 
  the 
  second 
  order, 
  will 
  contain 
  two 
  arbi- 
  

   trary 
  constants, 
  and 
  will 
  thus 
  render 
  the 
  solution 
  of 
  the 
  given 
  equation 
  complete, 
  

   without 
  the 
  introduction 
  of 
  any 
  other 
  terms. 
  

  

  Ex. 
  5. 
  (l-^)^-(2^ 
  + 
  l>^-K-^)«=0. 
  

  

  This 
  equation 
  has 
  been 
  solved 
  by 
  Professor 
  Boole 
  ; 
  but 
  it 
  is 
  requisite 
  to 
  shew 
  

   that 
  his 
  solution 
  is 
  not 
  confined 
  to 
  integral 
  values 
  of 
  m. 
  

   The 
  symbolical 
  form 
  of 
  the 
  equation 
  is 
  

  

  D(D-l) 
  • 
  

  

  Let 
  u=e- 
  mS 
  f(-T$)v\ 
  then 
  

  

  /^ 
  D 
  )- 
  ( 
  D 
  - 
  w 
  ,)(D-^-i 
  y^ 
  D 
  " 
  2 
  ) 
  g2 
  ^ 
  =0 
  - 
  

  

  By 
  making 
  /(^ 
  ^^.^ 
  (-D^i 
  

  

  or 
  /(_D)=t£i2 
  

  

  this 
  equation 
  is 
  reduced 
  to 
  

  

  (D-2) 
  2 
  -? 
  2 
  2 
  , 
  A/1 
  , 
  

  

  "~Ty(L^Tr 
  e 
  -'«=o(i) 
  

  

  and 
  M=e 
  -^L=-« 
  =( 
  _i)-^ 
  r 
  

  

  = 
  I 
  — 
  J 
  | 
  A 
  cos 
  (g 
  sin 
  ~~ 
  1 
  x) 
  + 
  B 
  sin 
  (^ 
  sin 
  ~ 
  1 
  #) 
  I 
  

  

  This 
  equation 
  is, 
  however, 
  susceptible 
  of 
  a 
  solution 
  in 
  a 
  different 
  form 
  from 
  

   that 
  which 
  we 
  have 
  just 
  exhibited. 
  

  

  Let 
  u 
  = 
  / 
  ( 
  — 
  2) 
  v 
  > 
  tnen 
  

  

  f[—2) 
  ■ 
  F^b) 
  L 
  VC--g- 
  + 
  l> 
  — 
  

  

  VOL. 
  XX. 
  PART 
  I. 
  O 
  

  

  