﻿along 
  a 
  Periodically 
  Loaded 
  String. 
  

  

  357 
  

  

  

  Fi 
  

  

  ET.1. 
  

  

  B 
  

  

  

  B 
  2 
  B,_! 
  B 
  r 
  

  

  AAA 
  A 
  

  

  = 
  — 
  00 
  X 
  — 
  

  

  

  

  For 
  the 
  string 
  B 
  r 
  _xB 
  r 
  

  

  

  

  #=. 
  + 
  00 
  

  

  f 
  r 
  =(p 
  r 
  cos 
  -^ 
  + 
  Q 
  r 
  sin 
  25: 
  U 
  

  

  n< 
  

  

  (ii.) 
  

  

  Now 
  at 
  t 
  r 
  r 
  = 
  0, 
  f 
  r 
  =y 
  r 
  _ 
  1; 
  and 
  at 
  %~ 
  = 
  l, 
  %r—y 
  r 
  ; 
  

  

  Q^'«'= 
  

  

  (iii.) 
  

  

  sin 
  — 
  

  

  The 
  equation 
  of 
  motion 
  of 
  B 
  r 
  , 
  for 
  r>0, 
  is 
  

  

  M^+2ty;=T 
  j 
  (|£±!) 
  - 
  (|^) 
  , 
  |. 
  . 
  (iv.) 
  

  

  Siibstituting-from 
  (ii.) 
  and 
  (iii.) 
  and 
  remembering 
  that 
  y 
  r 
  <* 
  e 
  tH 
  ', 
  

  

  v 
  r 
  (- 
  Mn 
  2 
  + 
  2kin) 
  = 
  — 
  ( 
  Q 
  r+ 
  , 
  + 
  P 
  r 
  sin 
  - 
  - 
  Q 
  r 
  cos 
  - 
  V 
  1 
  

  

  v 
  V 
  u 
  v 
  / 
  

  

  Tn 
  / 
  nZ 
  \ 
  

  

  = 
  — 
  — 
  ^y 
  r+l 
  -2y 
  r 
  cos- 
  + 
  ?6-iJ, 
  

  

  or 
  

  

  « 
  

  

  . 
  nZ 
  

   7 
  vsin 
  — 
  

  

  yr+i+j/r-i 
  — 
  2^ 
  I 
  cos— 
  H 
  g^-2X:nf 
  — 
  M>tM 
  =0. 
  . 
  (v.) 
  

  

  The 
  equation 
  of 
  motion 
  of 
  B 
  is 
  

  

  This 
  gives 
  

  

  y 
  (-Mn 
  2 
  + 
  2hu) 
  =(~ 
  Q, 
  + 
  T 
  -^- 
  (A-B))^' 
  

  

  Tn 
  / 
  nl\ 
  Tin 
  . 
  . 
  _,. 
  . 
  . 
  , 
  .. 
  

  

  = 
  -7-^ 
  ^i-i/o 
  cos 
  ~J 
  + 
  — 
  (A-B>"". 
  . 
  (vn.) 
  

  

  v 
  sin 
  — 
  

  

  