﻿432 
  Mr. 
  J. 
  Hollingworth 
  on 
  a 
  Physical 
  

  

  Now 
  for 
  present 
  considerations 
  X 
  represents 
  a 
  fixed 
  point 
  

   on 
  the 
  wire, 
  and 
  therefore 
  X 
  is 
  a 
  constant. 
  

  

  Put 
  /. 
  KXforK, 
  K(X 
  + 
  dX) 
  for 
  K', 
  

  

  GX 
  „ 
  G, 
  G(X 
  + 
  dX) 
  „ 
  G', 
  

  

  5.,, 
  r, 
  n/(x+dX) 
  „ 
  B', 
  

  

  ^ 
  „ 
  L, 
  L/(X 
  + 
  ^X) 
  „ 
  L'. 
  

  

  Then 
  the 
  new 
  values 
  of 
  K 
  L 
  R 
  G 
  are 
  all 
  constants 
  if 
  dX 
  is 
  

   specified 
  beforehand. 
  

  

  Also, 
  referring 
  to 
  equation 
  (3), 
  we 
  get 
  

  

  (KV+G') 
  (!/*> 
  + 
  R') 
  

  

  _ 
  (Kip 
  + 
  G) 
  (X 
  + 
  dX) 
  {Up 
  + 
  R 
  ) 
  

  

  X 
  + 
  dX 
  

   = 
  (Ktp 
  + 
  G-)(Li> 
  + 
  R), 
  

   .*. 
  in 
  this 
  case 
  a 
  = 
  a', 
  

  

  i. 
  e. 
  the 
  velocity 
  of 
  propagation 
  of 
  the 
  disturbance 
  is 
  the 
  same 
  

   in 
  both 
  portions 
  of 
  the 
  wire. 
  

  

  Equations 
  (1) 
  and 
  (2) 
  now 
  become 
  

  

  Ae- 
  iaX 
  + 
  A 
  / 
  e 
  iaX 
  = 
  A 
  1 
  e~ 
  iaX 
  , 
  (4) 
  

  

  X(Kip 
  + 
  G){Ae- 
  iaX 
  -A'e 
  iaX 
  ) 
  = 
  A 
  1 
  e- 
  iaX 
  (X 
  + 
  dX)(Ki 
  P 
  A-G), 
  

   i. 
  e. 
  Ae- 
  iaX 
  -A'e 
  iaX 
  =A 
  1 
  e- 
  iaX 
  . 
  (l+ 
  ~ 
  \, 
  . 
  (5) 
  

  

  solving 
  (4) 
  and 
  (5) 
  for 
  A' 
  and 
  A^ 
  in 
  terms 
  of 
  A 
  

  

  a;=-a*-**||, 
  (6) 
  

  

  A 
  - 
  A 
  (^S> 
  • 
  <?. 
  

  

  Hence 
  when 
  the 
  forward 
  wave 
  reaches 
  the 
  point 
  X, 
  a 
  wave 
  

   of 
  amplitude 
  Ae~ 
  2iaX 
  ^ 
  is 
  reflected 
  back, 
  and 
  one 
  of 
  ampli- 
  

   tude 
  A( 
  1— 
  ^-= 
  | 
  goes 
  forward. 
  

  

  Of 
  course, 
  throughout 
  the 
  whole 
  calculation 
  only 
  the 
  real 
  

   portion 
  of 
  the 
  original 
  expression 
  

  

  & 
  e 
  i(pt±ax) 
  i 
  s 
  taken, 
  

   and 
  hence, 
  since 
  a 
  is 
  complex, 
  A 
  is 
  also 
  complex, 
  and 
  

  

  