﻿IN 
  THE 
  DIFFERENTIAL 
  CALCULUS. 
  43 
  

  

  1 
  -2 
  + 
  rfx 
  

  

  I 
  _r_ 
  f 
  1 
  2 
  _ 
  r 
  r 
  + 
  2 
  « 
  V+ 
  2 
  -^ 
  

  

  j 
  2 
  11.2* 
  r 
  + 
  2-rjU 
  / 
  r 
  + 
  2 
  

  

  ^r^r+8-r^ 
  (r 
  + 
  2) 
  (2 
  r 
  + 
  2) 
  1 
  

  

  /2r 
  + 
  2 
  (r 
  + 
  2-r 
  i 
  u)(2/ 
  + 
  2-/ 
  j 
  u) 
  + 
  J 
  

  

  j 
  -2 
  + 
  rfX 
  

   d 
  r 
  ~ 
  X 
  I 
  r**y\ 
  I 
  r 
  r 
  ( 
  z 
  2 
  , 
  a 
  r 
  x 
  r+2 
  r 
  + 
  2 
  \ 
  

  

  J7~-A 
  X 
  ^)-~r^~ 
  a 
  \TT2 
  + 
  ^2r- 
  + 
  -2^r-f 
  J 
  . 
  +kc 
  -) 
  

  

  l~r 
  

   = 
  aV> 
  (B). 
  

  

  The 
  same 
  process 
  will 
  give 
  the 
  same 
  result 
  for 
  all 
  the 
  terms 
  of 
  this 
  expan- 
  

   sion 
  except 
  the 
  last. 
  But 
  as 
  ' 
  ~t^ 
  is 
  a 
  factor 
  of 
  that 
  term, 
  it 
  is 
  evidently 
  0. 
  

  

  Consequently, 
  the 
  equation 
  — 
  — 
  ; 
  (* 
  r/ 
  *T^ 
  ) 
  — 
  ax 
  r 
  ^y 
  gives 
  the 
  complete 
  value 
  

  

  dx 
  r 
  ' 
  \ 
  ax/ 
  

  

  of 
  y=d' 
  r 
  l 
  e 
  ax 
  , 
  it 
  being 
  observed 
  that 
  one 
  of 
  the 
  constants 
  of 
  integration 
  is 
  0. 
  The 
  

   other 
  constants 
  must 
  be 
  determined 
  by 
  means 
  of 
  equation 
  (A). 
  

  

  10. 
  To 
  find 
  d* 
  cos 
  a 
  x. 
  

  

  a 
  2 
  a 
  4 
  I 
  

  

  Since 
  cos 
  ax=\—^ 
  + 
  u 
  xi 
  - 
  &c. 
  [ 
  

  

  if 
  -« 
  2 
  = 
  b, 
  we 
  shall 
  have 
  

  

  l 
  -2+rp 
  

  

  (-L-'YcoBaz^-ryl* 
  ! 
  _!L 
  x*->-» 
  

   \ 
  x 
  r-i 
  dx) 
  K 
  ' 
  \\2 
  / 
  2 
  

  

  / 
  — 
  4 
  + 
  r/x 
  /-6 
  + 
  r 
  fx 
  

  

  + 
  /4 
  ri 
  + 
  /e 
  i~e 
  x 
  + 
  '■■ 
  

  

  ) 
  

  

  I 
  r 
  . 
  j 
  r 
  

  

  It 
  remains 
  to 
  express 
  this 
  series 
  in 
  the 
  form 
  of 
  a 
  differential 
  equation 
  when 
  

  

  r 
  is 
  a 
  whole 
  number. 
  

  

  /— 
  2 
  s 
  + 
  r 
  jjl 
  

  

  I 
  

  

  Let 
  s 
  be 
  any 
  whole 
  number 
  less 
  than 
  r+1, 
  S 
  = 
  (-r)^6 
  s 
  '- 
  — 
  =L= 
  — 
  ; 
  then 
  the 
  

  

  i~T 
  

   series 
  may 
  be 
  written 
  

  

  ff 
  - 
  S 
  f 
  *"~ 
  r 
  * 
  , 
  * 
  (r 
  + 
  2s)(2r 
  + 
  2s) 
  r 
  2r+2,- 
  r 
  . 
  .) 
  

  

  V 
  ~ 
  \ 
  j2_s 
  + 
  /2r 
  + 
  2 
  t 
  s 
  (r 
  + 
  2s-rfM){2r 
  + 
  2s-rfi) 
  + 
  <vc. 
  j 
  

  

  rf*- 
  1 
  / 
  rf, 
  dy\ 
  (2_s-rjx 
  x^~ 
  r 
  * 
  r 
  - 
  2s+r 
  

  

  ^""^Hl 
  

  

  ' 
  dx 
  r 
  ~ 
  l 
  \ 
  dx) 
  \ 
  2s 
  i 
  2s— 
  r 
  /2 
  s 
  + 
  r—1 
  r 
  + 
  2 
  s 
  — 
  r 
  fx 
  

  

  VOL. 
  XX. 
  PART 
  I. 
  M 
  

  

  