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  II. 
  — 
  On 
  a 
  Process 
  in 
  the 
  Differential 
  Calculus, 
  and 
  its 
  application 
  to 
  the 
  Solu- 
  

   tion 
  of 
  certain 
  Differential 
  Equations. 
  By 
  the 
  Rev. 
  P. 
  Kelland, 
  M.A., 
  

   F.R.SS.L.fyE., 
  F.C.P.S., 
  late 
  Fellow 
  of 
  Queen's 
  College, 
  Cambridge; 
  Professor 
  

   of 
  Mathematics, 
  fyc, 
  in 
  the 
  University 
  of 
  Edinburgh. 
  

  

  (Read 
  17th 
  December 
  1849.) 
  

  

  The 
  facilities 
  which 
  are 
  afforded 
  by 
  the 
  introduction 
  of 
  the 
  function 
  / 
  — 
  into 
  

   certain 
  classes 
  of 
  Differential 
  expressions 
  are 
  well 
  known. 
  This 
  function 
  has 
  

   effected 
  the 
  combination 
  and 
  generalization 
  of 
  Problems, 
  which, 
  although 
  found 
  

   to 
  be 
  capable 
  of 
  solution 
  in 
  particular 
  cases, 
  were 
  regarded 
  rather 
  as 
  isolated 
  and 
  

   exceptional 
  forms, 
  than 
  as 
  integral 
  parts 
  of 
  some 
  comprehensive 
  expression. 
  

   But 
  the 
  subject 
  is 
  far 
  from 
  exhausted. 
  Some 
  of 
  the 
  most 
  important 
  Differential 
  

   Equations 
  have 
  never 
  as 
  yet 
  been 
  solved 
  by 
  a 
  general 
  method. 
  The 
  present 
  

   Memoir 
  is 
  intended 
  to 
  supply 
  this 
  defect. 
  The 
  process 
  employed 
  differs 
  little 
  

   from 
  that 
  which 
  I 
  have 
  previously 
  exhibited 
  ; 
  but 
  the 
  range 
  of 
  Problems 
  which 
  

   it 
  embraces 
  is 
  much 
  more 
  extensive, 
  and 
  the 
  Problems 
  themselves 
  are 
  of 
  a 
  

   more 
  important 
  character. 
  

  

  Section 
  I. 
  Preliminary 
  Theorems. 
  

  

  1. 
  Let 
  y=e~°- 
  xr 
  , 
  

  

  then 
  [A-l^y^ary, 
  ; 
  

  

  from 
  which 
  it 
  follows, 
  since 
  the 
  operation 
  reproduces 
  the 
  function 
  itself, 
  that 
  

  

  (tti 
  Jl) 
  * 
  ' 
  y={- 
  ar 
  Ty 
  (!•)» 
  whatever 
  fx 
  may 
  be. 
  

  

  It 
  is 
  necessary 
  to 
  observe, 
  that 
  if 
  fx 
  be 
  negative, 
  the 
  above 
  equation 
  takes 
  no 
  

   cognizance 
  of 
  functions 
  of 
  integration, 
  which 
  would 
  be 
  introduced 
  by 
  means 
  of 
  

   the 
  added 
  arbitrary 
  constants 
  ; 
  and 
  this 
  remark 
  applies 
  to 
  all 
  our 
  processes. 
  

  

  In 
  the 
  equation 
  / 
  e~ 
  6 
  6 
  n 
  ~ 
  l 
  d6=jn, 
  

   let 
  d=az 
  r 
  > 
  

  

  then 
  f*e- 
  m 
  ' 
  r 
  a 
  n 
  - 
  1 
  ll 
  r(»-V 
  1 
  ?dq=in 
  

  

  or 
  

  

  VOL. 
  XX. 
  PART 
  I. 
  

  

  J?-= 
  f™ 
  a"' 
  1 
  e—" 
  r 
  da, 
  

  

  x 
  rn 
  Jo 
  

  

  