﻿434: 
  Mr. 
  J. 
  Hcllingworth 
  on 
  a 
  Pltyslcal 
  

  

  The 
  above 
  being 
  the 
  case 
  when 
  waves 
  in 
  both 
  directions 
  

   are 
  originally 
  incident. 
  

  

  Now 
  at 
  a 
  point 
  dX 
  further 
  on 
  let 
  the 
  constants 
  of 
  the 
  wire 
  

   change 
  again 
  

  

  from 
  K(X 
  + 
  dX) 
  to 
  K(X 
  + 
  2dX) 
  and 
  similarly. 
  

  

  Another 
  series 
  of 
  reflexions 
  and 
  refractions, 
  similar 
  to 
  those 
  

   already 
  calculated, 
  occur. 
  

  

  Repeating 
  this 
  process 
  we 
  can 
  divide 
  the 
  length 
  of 
  wire 
  

   between 
  x 
  = 
  ot 
  and 
  a?=yS 
  into 
  n 
  equal 
  portions 
  dX, 
  and 
  then 
  

   the 
  following 
  relations 
  will 
  hold 
  between 
  the 
  amplitudes 
  of 
  

   the 
  waves 
  in 
  successive 
  portions 
  : 
  — 
  

  

  A 
  A 
  s 
  dX 
  B 
  s 
  e 
  2iaX 
  dX 
  

  

  As+1 
  ~ 
  As 
  ~ 
  ~2XT 
  + 
  " 
  ~2X 
  ' 
  

  

  B 
  -B-l,+ 
  — 
  "^— 
  -A 
  r-™* 
  dX 
  

  

  ^i-^+it 
  ^-vt- 
  J\ 
  s 
  e 
  ^, 
  

  

  . 
  . 
  dX 
  ^ 
  e 
  2iaX 
  dX 
  ,.-_. 
  

  

  A 
  s 
  +i— 
  A 
  s 
  — 
  — 
  Aj^r 
  +J3 
  S 
  ^y 
  — 
  9 
  • 
  (12) 
  

  

  B.-B^i= 
  -^ 
  A 
  °—2lT~-> 
  • 
  ( 
  13 
  ) 
  

  

  (12) 
  and 
  (13) 
  being 
  in 
  sequence 
  form, 
  we 
  can, 
  by 
  addition, 
  

   find 
  the 
  relation 
  between 
  any 
  pair 
  of 
  A's 
  and 
  any 
  pair 
  of 
  B's. 
  

   Now 
  let 
  n 
  become 
  infinitely 
  large, 
  and 
  therefore 
  dX 
  small, 
  

   and 
  effect 
  the 
  summation 
  from 
  x 
  = 
  a 
  to 
  x 
  = 
  x, 
  (12) 
  and 
  (13) 
  

   then 
  give 
  

  

  a 
  - 
  a 
  - 
  - 
  (' 
  *#* 
  j. 
  f 
  ' 
  r 
  e 
  ' 
  i<atd 
  Z 
  n 
  tf 
  

  

  A 
  * 
  A 
  -- 
  J..^r 
  + 
  J. 
  B 
  '."^r*- 
  • 
  (u) 
  

  

  and 
  for 
  the 
  B's 
  

  

  e-^tdi; 
  

  

  B,-B. 
  = 
  f^_fA 
  f 
  

  

  (15) 
  

  

  A 
  a 
  B 
  a 
  being 
  the 
  amplitudes 
  of 
  the 
  forward 
  and 
  reflected 
  

   waves 
  at 
  the 
  point 
  x 
  = 
  a, 
  and 
  therefore 
  constants 
  for 
  the 
  

   particular 
  value 
  of 
  x 
  involved. 
  

  

  Hence, 
  if 
  we 
  can 
  solve 
  equations 
  (14) 
  and 
  (15) 
  for 
  A 
  x 
  and 
  

   B 
  z 
  , 
  we 
  have 
  the 
  amplitudes 
  of 
  the 
  forward 
  and 
  reflected 
  

   waves 
  at 
  any 
  point 
  x 
  on 
  the 
  wire 
  for 
  which 
  

  

  K 
  = 
  K> 
  L= 
  &c, 
  

  

  i. 
  e. 
  for 
  the 
  disk 
  considered 
  on 
  p. 
  429. 
  

  

  