﻿• 
  r 
  >4 
  PROFESSOR 
  KELLAND 
  ON 
  A 
  PROCESS 
  

  

  Ex. 
  2. 
  To 
  solve 
  the 
  equation 
  

  

  *1 
  - 
  a 
  diy 
  ly 
  =X 
  

  

  dx 
  d 
  x 
  \ 
  2x 
  

  

  ] 
  . 
  By 
  the 
  method 
  employed 
  at 
  p. 
  269 
  of 
  my 
  previous 
  Memoir, 
  this 
  equation 
  

   becomes 
  

  

  (-D 
  + 
  *)y 
  + 
  a 
  (-l)*(=5±l^=-Xz 
  

   /-D+i 
  

  

  « 
  rf-* 
  y 
  1 
  X 
  

  

  Or 
  - 
  — 
  a 
  — 
  

  

  z 
  

  

  x 
  dx~% 
  * 
  x 
  D 
  — 
  % 
  

  

  j_ 
  rxdx 
  

  

  ~ 
  si* 
  J 
  a;* 
  

   =P, 
  suppose 
  ; 
  

  

  , 
  , 
  y 
  1 
  Id 
  P 
  tfi 
  p\ 
  

  

  then 
  -=-j 
  (tt 
  - 
  + 
  a 
  — 
  r) 
  

  

  a- 
  ux 
  

  

  dx 
  

  

  A 
  a 
  2 
  * 
  «»* 
  /* 
  -« 
  2j: 
  . 
  (d 
  P 
  rf*P\ 
  

  

  y 
  = 
  Axe 
  +xe 
  I 
  e 
  dx\-— 
  - 
  + 
  a 
  — 
  r 
  | 
  

   J 
  \dx 
  dx 
  i) 
  

  

  2. 
  By 
  Professor 
  Boole's 
  method, 
  given 
  in 
  the 
  Philosophical 
  Magazine 
  for 
  

   February 
  1847. 
  The 
  given 
  equation 
  when 
  written 
  

  

  x 
  {d— 
  a 
  d*)y—jty=H 
  x 
  ; 
  

   may 
  be 
  thrown 
  into 
  the 
  form 
  

  

  / 
  (d) 
  (x 
  F 
  (of) 
  ) 
  y 
  = 
  X 
  x 
  ; 
  provided 
  

  

  /(d)F(d) 
  = 
  d-ad 
  } 
  , 
  and 
  

   f'(d)F(d)=-h 
  

  

  We 
  have, 
  therefore, 
  / 
  (d) 
  = 
  (d 
  } 
  -a)~ 
  l 
  

  

  F(d) 
  = 
  (di-a) 
  2 
  di 
  

   and 
  y 
  = 
  d-* 
  (d±-a)- 
  2 
  {x- 
  1 
  (di-a)Xx\ 
  

  

  3. 
  By 
  Mr 
  Hargreave's 
  method, 
  given 
  in 
  the 
  Philosophical 
  Transactions 
  for 
  

   1848, 
  p. 
  31. 
  

  

  By 
  changing 
  d 
  into 
  x, 
  and 
  x 
  into 
  — 
  d 
  in 
  the 
  equation 
  

  

  x 
  (d—a 
  rf*) 
  j/-jy 
  = 
  Xa; 
  .... 
  (1.) 
  

   it 
  becomes 
  d(x—az$)y 
  + 
  iy=dX 
  . 
  . 
  . 
  . 
  (2.) 
  

  

  which, 
  being 
  an 
  ordinary 
  linear 
  equation, 
  gives, 
  as 
  its 
  solution, 
  if 
  we 
  write 
  

  

  d~ 
  l 
  for 
  dx, 
  

  

  y 
  = 
  x-* 
  (x> 
  -a)' 
  2 
  d- 
  l 
  {(zi-a) 
  dX] 
  .... 
  (3.) 
  

  

  Now, 
  since 
  equation 
  (2.) 
  has 
  been 
  derived 
  from 
  equation 
  (1.) 
  by 
  the 
  change 
  

   of 
  d 
  into 
  x, 
  and 
  x 
  into 
  — 
  d, 
  equation 
  (1.) 
  may 
  be 
  derived 
  from 
  equation 
  (2.) 
  by 
  

   changing 
  x 
  into 
  d 
  and 
  d 
  into 
  —x. 
  Consequently 
  (in 
  some 
  cases, 
  at 
  least, 
  of 
  

   which 
  this 
  form 
  is 
  an 
  instance) 
  the 
  solution 
  of 
  equation 
  (1.) 
  may 
  be 
  derived 
  from 
  

   the 
  solution 
  of 
  equation 
  (2.) 
  by 
  the 
  same 
  change. 
  Hence 
  the 
  solution 
  of 
  the 
  given 
  

   equation 
  is 
  

  

  