﻿40 
  PROFESSOR 
  KELLAND 
  ON 
  A 
  PROCESS 
  

  

  or, 
  which 
  is 
  the 
  same 
  thing, 
  

  

  

  r 
  

  

  Jo 
  " 
  r 
  e 
  * 
  da 
  > 
  from 
  wni 
  ch 
  it 
  follows, 
  by 
  equation 
  (1.), 
  that 
  

  

  h 
  

  

  I 
  

  

  fa 
  ( 
  1 
  d\/* 
  1 
  r™ 
  7- 
  1 
  

  

  2±Iff_ 
  l 
  

  

  \m 
  + 
  r 
  /j. 
  

   . 
  s* 
  I 
  r 
  

  

  x 
  m 
  + 
  rfj, 
  

  

  \m 
  + 
  r 
  fj. 
  

  

  Consequently, 
  (-K 
  £-Y-- 
  = 
  (-rfl—L— 
  — 
  i— 
  . 
  . 
  . 
  (2.) 
  

  

  Let 
  u 
  be 
  a 
  function, 
  supposed 
  to 
  be 
  capable 
  of 
  being 
  represented 
  under 
  the 
  

   form 
  of 
  a 
  series 
  of 
  functions, 
  such 
  as 
  e- 
  a 
  * 
  ; 
  then 
  we 
  shall 
  have 
  

  

  * 
  fc=T3i) 
  M 
  =(-^ 
  "-py— 
  • 
  • 
  • 
  ■ 
  <N 
  

   2. 
  Since 
  — 
  j 
  j-» 
  = 
  ,_i 
  . 
  =-yr 
  

  

  a; 
  r 
  L 
  dx 
  x 
  r 
  d 
  x 
  dx 
  

  

  it 
  follows 
  that 
  (— 
  -r 
  -r- 
  V 
  «=»•'* 
  (-—V 
  «« 
  

  

  Va^- 
  1 
  dx) 
  \dx 
  r 
  ) 
  

  

  + 
  u 
  

  

  u.r 
  ( 
  d 
  \P. 
  , 
  ,/ 
  / 
  

  

  3. 
  If 
  we 
  write 
  dL 
  w 
  for 
  — 
  37 
  3- 
  M 
  ; 
  then 
  from 
  the 
  last 
  Art. 
  it 
  will 
  be 
  evident, 
  

  

  x 
  r 
  ax 
  

  

  without 
  demonstration, 
  that 
  

  

  d^ 
  (u 
  v) 
  = 
  v 
  d 
  r 
  u 
  + 
  /jl 
  d 
  r 
  v 
  a 
  r 
  u 
  + 
  . 
  2 
  ^ 
  r 
  v< 
  * 
  r 
  u 
  + 
  ^ 
  c 
  ' 
  

  

  4. 
  Let 
  <p 
  (d 
  r 
  ) 
  be 
  any 
  function 
  of 
  d 
  r 
  capable 
  of 
  expansion 
  in 
  the 
  form 
  

   (p 
  (d 
  r 
  ) 
  = 
  2 
  ad/, 
  then, 
  by 
  the 
  last 
  Article 
  

  

  d) 
  (d 
  r 
  ) 
  (uv) 
  = 
  1 
  a 
  {v 
  d^u 
  + 
  fxd 
  r 
  v 
  d^~ 
  w 
  + 
  &c} 
  

  

  = 
  v(2ad' 
  A 
  )u 
  + 
  d 
  v(2afld\.~ 
  )u 
  + 
  kc. 
  

  

  