﻿IN 
  THE 
  DIFFERENTIAL 
  CALCULUS. 
  41 
  

  

  = 
  v(p(d 
  r 
  )u 
  + 
  d 
  r 
  v(p'(d 
  r 
  )u 
  + 
  ^— 
  ^ 
  d% 
  v 
  (p" 
  (d 
  r 
  )u 
  + 
  &c. 
  

  

  where 
  <p' 
  (d 
  r 
  ) 
  is 
  the 
  differential 
  coefficient 
  of 
  cp 
  (d 
  r 
  ) 
  with 
  respect 
  to 
  d 
  r 
  . 
  

  

  5. 
  In 
  Art. 
  3, 
  let 
  «=/*\ 
  then 
  

  

  14 
  /S 
  * 
  r 
  (ix 
  r 
  p 
  _ 
  p 
  — 
  1 
  

  

  d 
  (e 
  u)= 
  e 
  {d 
  u 
  + 
  /a 
  p 
  r 
  d 
  r 
  u 
  

  

  + 
  ^-l) 
  (/3r)2 
  ^-2 
  M 
  + 
  &c<} 
  

  

  = 
  f* 
  (d 
  r 
  + 
  (3rfu; 
  

   and 
  hence, 
  generally, 
  as 
  in 
  Art. 
  1., 
  

  

  <p(d 
  r 
  ){e 
  lixr 
  u} 
  = 
  e 
  lixr 
  <t>(d 
  r 
  + 
  Pr)u 
  (5.) 
  

  

  6. 
  Let 
  0(z)=2B/* 
  r 
  , 
  then 
  

  

  (a;) 
  +(^)« 
  = 
  2B 
  e 
  fi 
  ** 
  ^ 
  (d 
  r 
  ) 
  u 
  

  

  = 
  2 
  B 
  ^ 
  (rf 
  r 
  - 
  /3 
  r) 
  e^ 
  r 
  « 
  by 
  (5) 
  

  

  =2B{^(^)-^K)^^ 
  + 
  ^"K)x^ 
  2 
  -&c.}/'^ 
  

  

  = 
  -4/ 
  (rf 
  r 
  ) 
  {2B 
  e* 
  3 
  *'«} 
  - 
  ^' 
  (d 
  r 
  ) 
  R 
  (2 
  B 
  e' 
  3 
  *>} 
  + 
  &c. 
  

   = 
  4 
  Vr) 
  {$ 
  CO 
  «*} 
  - 
  -4/ 
  (rf 
  r 
  ) 
  K 
  (*) 
  . 
  "} 
  

   + 
  =— 
  s 
  •¥ 
  (d 
  r 
  ) 
  {d* 
  (*) 
  . 
  u] 
  — 
  &c. 
  

  

  7. 
  It 
  is 
  easily 
  seen 
  that, 
  if 
  m 
  be 
  a 
  whole 
  number, 
  

  

  -m 
  r~ 
  m 
  x 
  rm 
  f 
  1 
  /l 
  1 
  „ 
  1\ 
  ) 
  

  

  and 
  that 
  in 
  other 
  cases, 
  

  

  d 
  r 
  log 
  x 
  = 
  (-rf 
  [fix 
  

  

  — 
  w 
  , 
  / 
  t- 
  — 
  1 
  /"* 
  \ 
  m 
  

  

  8. 
  The 
  operation 
  denoted 
  by 
  d 
  r 
  must 
  not 
  be 
  confounded 
  with 
  \x 
  l 
  dx 
  ) 
  '• 
  

  

  it 
  is, 
  in 
  reality, 
  ( 
  Jx~ 
  dx\ 
  m 
  . 
  The 
  one 
  expression 
  can 
  be 
  deduced 
  from 
  the 
  

   other, 
  thus 
  : 
  — 
  

  

  (r-'i- 
  C 
  \ 
  m 
  i 
  r-l 
  ,-m 
  -(r-1) 
  , 
  r-l.-nl, 
  

  

  x 
  ldx\ 
  logx=x 
  d 
  x 
  v 
  J 
  \ogx 
  = 
  x 
  d 
  - 
  (; 
  

  

  - 
  (x 
  p 
  ~ 
  r+l 
  —- 
  9 
  ~ 
  r+1 
  > 
  

   P 
  

  

  I 
  /p—r 
  + 
  1 
  , 
  \ 
  / 
  /q—r 
  + 
  1 
  , 
  \ 
  

  

  .c-)-rH 
  ^ 
  +B, 
  L 
  H 
  * 
  + 
  "J,. 
  

  

  X 
  — 
  , 
  X 
  > 
  x 
  

  

  P 
  ! 
  / 
  p—r+l 
  j 
  q— 
  r+1 
  

  

  where 
  p, 
  q, 
  and 
  - 
  are 
  indefinitely 
  small. 
  

  

  