﻿52 
  PROFESSOR 
  KELLAND 
  ON 
  A 
  PROCESS 
  

  

  By 
  making 
  /(-| 
  + 
  D 
  = 
  p^"^/ 
  (-f 
  ) 
  • 
  • 
  • 
  00 
  

  

  this 
  equation 
  is 
  reduced 
  to 
  the 
  equation 
  of 
  the 
  first 
  order. 
  

  

  D 
  + 
  ra-2-? 
  24 
  / 
  D\ 
  -i 
  

  

  D 
  e 
  V 
  = 
  f 
  {—2) 
  ° 
  ■ 
  • 
  - 
  (2 
  - 
  } 
  

  

  V 
  — 
  

  

  Also 
  

  

  /H7 
  

  

  D\ 
  / 
  2 
  + 
  2 
  

  

  '(-f^H* 
  

  

  2 
  2 
  2 
  

  

  D 
  w 
  1 
  o 
  

  

  1 
  |-i- 
  

  

  _ 
  -(m-2+ 
  g 
  )o 
  ,' 
  2 
  2 
  2 
  2 
  ( 
  m 
  _2 
  + 
  ? 
  )(> 
  

  

  (m+2-l) 
  ^/l 
  df\«»+?-l 
  m 
  + 
  7-2 
  

  

  m= 
  ( 
  _ 
  2r 
  c 
  +J 
  -i) 
  /l 
  rf\ 
  

  

  which 
  is 
  of 
  the 
  ordinary 
  form 
  when 
  m 
  + 
  q 
  is 
  a 
  whole 
  number. 
  

   The 
  same 
  is 
  true 
  if 
  — 
  q 
  be 
  substituted 
  for 
  q. 
  

  

  Section 
  III. 
  Comparison 
  of 
  Processes. 
  

  

  We 
  shall 
  occupy 
  this 
  Section 
  in 
  the 
  comparison 
  of 
  the 
  different 
  methods 
  

   which 
  may 
  be 
  employed 
  in 
  the 
  solution 
  of 
  a 
  given 
  Differential 
  Equation. 
  

   13. 
  To 
  find 
  the 
  value 
  ofj-D 
  . 
  v. 
  

   Suppose 
  v 
  expressed 
  in 
  the 
  form 
  

  

  A 
  B 
  . 
  

  

  v 
  = 
  — 
  + 
  — 
  — 
  + 
  OIC. 
  

  

  then 
  f-D 
  . 
  v=A 
  lne- 
  n 
  * 
  + 
  Bf>^e-( 
  n 
  + 
  r 
  )< 
  + 
  &c. 
  

  

  . 
  fn 
  -nln 
  + 
  r 
  

   = 
  A- 
  — 
  + 
  B' 
  — 
  — 
  +&c. 
  

   x 
  n 
  x 
  n 
  + 
  r 
  

  

  /—ax 
  , 
  , 
  . 
  n 
  — 
  1 
  -,-. 
  n 
  + 
  r 
  — 
  1 
  n 
  

  

  e 
  d 
  a 
  (A 
  a 
  + 
  B 
  a 
  + 
  . 
  . 
  .) 
  

  

  / 
  —ax 
  GO. 
  ,. 
  n 
  -r>" 
  + 
  r 
  .o 
  

  

  = 
  / 
  e 
  (A 
  a 
  + 
  B 
  a 
  + 
  &c. 
  

  

  Hence, 
  if 
  «=/(*) 
  

  

  'F5/»-r^ 
  / 
  (I) 
  

  

  