﻿along 
  a 
  Periodically 
  Loaded 
  String. 
  

  

  361 
  

  

  From 
  what 
  has 
  been 
  already 
  said, 
  it 
  is 
  obvious 
  that 
  the 
  

   wave 
  will 
  penetrate 
  the 
  masses 
  if 
  yjr 
  lies 
  within 
  limits 
  cor- 
  

   responding 
  to 
  the 
  portions 
  A 
  C 
  , 
  AjCi, 
  A 
  2 
  C 
  2 
  , 
  &c. 
  of 
  the 
  

   curve. 
  If, 
  on 
  the 
  other 
  hand, 
  yjr 
  belongs 
  to 
  the 
  regions 
  

   C 
  A,, 
  CiA-j, 
  O2A3, 
  &c, 
  the 
  motion 
  will 
  only 
  enter 
  to 
  a 
  small 
  

   distance. 
  

  

  5. 
  It 
  is 
  interesting 
  to 
  look 
  at 
  the 
  magnitude 
  of 
  the 
  different 
  

   amplitudes 
  rather 
  more 
  closely. 
  We 
  shall 
  lose 
  no 
  generality 
  

   if 
  we 
  suppose 
  that 
  a 
  lies 
  between 
  and 
  — 
  "lir. 
  On 
  consider- 
  

   ing 
  the 
  signs 
  of 
  sin 
  a 
  and 
  cos 
  a 
  as 
  given 
  by 
  (ix.), 
  we 
  have 
  the 
  

   following 
  table 
  :— 
  

  

  Region. 
  

  

  ■ifi 
  between 
  

  

  sin 
  a. 
  

  

  z 
  and 
  cos 
  a. 
  

  

  a 
  between 
  

  

  ! 
  A 
  n 
  B 
  n 
  

  

  and 
  tt 
  

   and 
  7r 
  

   tt 
  and 
  2tt 
  

  

  tt 
  and 
  2-7T 
  

  

  + 
  

  

  + 
  

  

  + 
  

   + 
  

  

  and 
  — 
  jj-. 
  

   A 
  

  

  — 
  -r 
  and 
  — 
  tt. 
  

  

  A 
  

  

  3tt 
  

  

  — 
  tt 
  and 
  — 
  q-. 
  

  

  — 
  ~n 
  and 
  — 
  2rr. 
  

  

  B 
  n 
  A, 
  

  

  A,B, 
  

  

  B,A 
  

  

  

  Similar 
  limits 
  recur 
  for 
  the 
  other 
  reaches, 
  A 
  2 
  A 
  4 
  , 
  A 
  4 
  A 
  e 
  , 
  . 
  . 
  . 
  

  

  6. 
  Regions 
  for 
  which 
  z 
  2 
  >l. 
  

  

  From 
  the 
  above 
  table 
  it 
  appears 
  that 
  a. 
  is 
  equal 
  to 
  

   or 
  — 
  it 
  according 
  as 
  ^ 
  lies 
  between 
  2sit 
  and 
  (2s+l)7r, 
  or 
  

   between 
  (25 
  — 
  1) 
  it 
  and 
  2sir. 
  Now, 
  denoting 
  moduli 
  of 
  com- 
  

   plex 
  quantities 
  by 
  straight 
  brackets, 
  

  

  A 
  2 
  B 
  2 
  ^ 
  + 
  2^cos^+l 
  

  

  C 
  2 
  

  

  <J 
  S 
  

  

  4 
  sin 
  2 
  i/r 
  

   the 
  upper 
  or 
  lower 
  sign 
  being 
  taken 
  according 
  as 
  a 
  is 
  or 
  

  

  — 
  7T. 
  

  

  This 
  expression 
  is 
  equal 
  to 
  

  

  e? 
  cosh 
  /3 
  + 
  cos 
  ^r 
  

  

  sin 
  2 
  -yjr 
  

   But 
  + 
  cosh 
  ft 
  = 
  cos 
  ty 
  — 
  fju\jr 
  sin 
  yfr 
  ; 
  

  

  A 
  2 
  B 
  2 
  

  

  C 
  2 
  

  

  _ 
  e* 
  uyjr 
  

  

  = 
  + 
  77 
  

  

  (xvii.) 
  

  

  C 
  2 
  I 
  " 
  ' 
  2 
  sin^r' 
  ' 
  

   For 
  the 
  frequencies 
  which 
  are 
  not 
  propagated 
  we 
  thus 
  

  

  