﻿IN 
  THE 
  DIFFERENTIAL 
  CALCULUS. 
  55 
  

  

  y= 
  d-* 
  (S-a)-' 
  2 
  {x- 
  1 
  (<**-«) 
  a: 
  X}, 
  

   which 
  is 
  precisely 
  the 
  solution 
  given 
  by 
  Professor 
  Boole's 
  process. 
  

  

  We 
  shall 
  conclude 
  the 
  present 
  Memoir 
  by 
  comparing 
  the 
  methods 
  here 
  exhi- 
  

   bited 
  in 
  two 
  particular 
  cases 
  of 
  the 
  second 
  example. 
  

  

  Case 
  1. 
  Let 
  X=0; 
  then 
  the 
  first 
  method 
  gives 
  y 
  =Kxe 
  a 
  ' 
  x 
  \ 
  whilst 
  the 
  second 
  

   and 
  third 
  methods 
  give 
  

  

  y 
  = 
  d-^(dt-a)- 
  2 
  . 
  0. 
  

  

  Now, 
  in 
  a 
  former 
  paper 
  {Transactions 
  of 
  the 
  Royal 
  Society 
  of 
  Edinburgh, 
  

   vol. 
  xiv., 
  p. 
  252), 
  I 
  have 
  shewn, 
  in 
  Cor. 
  1. 
  to 
  Example 
  3, 
  that 
  

  

  (dt-a)- 
  2 
  . 
  = 
  Ae 
  a2 
  * 
  + 
  Bze 
  a2x 
  

   Hence 
  y=d-l{Ae 
  a2x 
  + 
  Bxe 
  a 
  **} 
  

  

  A 
  a 
  2 
  x 
  Br 
  a 
  2 
  x 
  B 
  a 
  2 
  x 
  

  

  = 
  —e 
  +— 
  e 
  -~—,e 
  

  

  a 
  a 
  la 
  

  

  The 
  condition 
  B=2 
  A 
  a 
  2 
  reduces 
  this 
  solution 
  to 
  the 
  former. 
  

   Case 
  2. 
  Let 
  X=b(—A-- 
  7 
  =- 
  -) 
  ; 
  

  

  \ 
  X* 
  V 
  <S 
  Xf 
  

  

  then, 
  by 
  the 
  first 
  method, 
  

  

  p 
  _ 
  J_ 
  f^dx_ 
  b 
  f 
  1 
  2«a/_ 
  : 
  

  

  sjxJ 
  sjx 
  s/x 
  \ 
  x 
  *Jn 
  

  

  1 
  2aV-ll_) 
  

   b 
  2ab\/^l 
  1 
  

  

  x? 
  sjv 
  x 
  

  

  dV 
  dtp 
  _ 
  /3 
  J_ 
  a 
  2 
  \ 
  

  

  lix- 
  + 
  a 
  d~xT~ 
  \2z>*x$) 
  

  

  a? 
  x 
  , 
  a 
  2 
  x 
  C 
  -a 
  2 
  x 
  , 
  /3 
  1 
  a 
  2 
  \ 
  

  

  <=Axe 
  a 
  + 
  bxe 
  I 
  e 
  dxi- 
  — 
  +— 
  ) 
  

  

  A 
  « 
  2 
  ^ 
  b 
  

  

  — 
  Kxe 
  r 
  

  

  s/x 
  

  

  By 
  the 
  second 
  and 
  third 
  methods, 
  

  

  - 
  (d 
  -a)Xx 
  = 
  b 
  [^-^ 
  - 
  2 
  - 
  7r 
  - 
  v 
  =^) 
  

   Also 
  d-*(rf*-a)- 
  8 
  = 
  ^ 
  + 
  2a 
  + 
  f* 
  

  

  QN 
  _2 
  , 
  / 
  3 
  a 
  2 
  a 
  4 
  \ 
  

   \ 
  4^ 
  ^ 
  a;*/ 
  

  

  an 
  ordinary 
  linear 
  differential 
  equation, 
  of 
  which 
  the 
  solution 
  is 
  

  

  Aa 
  2 
  x 
  , 
  -r> 
  a 
  2 
  x 
  

   xe 
  +Be 
  — 
  . 
  

  

  s/x 
  

  

  This 
  agrees 
  with 
  the 
  former 
  solution 
  by 
  making 
  B 
  = 
  0. 
  

   vol. 
  xx. 
  PART 
  I. 
  

  

  