﻿358 
  Mr. 
  C. 
  Godfrey 
  on 
  Discontinuities 
  of 
  Wave-motion 
  

  

  We 
  will 
  now 
  abbreviate 
  by 
  writing 
  

  

  nl 
  , 
  

   v 
  r 
  

  

  Mv 
  2 
  

   2/T 
  

  

  = 
  /x; 
  

  

  . 
  nl 
  

   . 
  v 
  sm 
  — 
  

   n 
  t 
  v 
  

  

  cos 
  1 
  ^r- 
  (2kni-~ 
  Mn 
  2 
  ) 
  = 
  coa(a 
  + 
  ifi) 
  = 
  cos 
  0, 
  . 
  (viii.) 
  

  

  where 
  a 
  and 
  /3 
  are 
  real. 
  

  

  , 
  _ 
  ??/ 
  Mwu 
  . 
  nZ 
  , 
  . 
  . 
  . 
  "1 
  

  

  cos 
  a 
  cosh 
  H 
  = 
  cos 
  ^rwr- 
  sin 
  — 
  =cos 
  vr— 
  ixxr 
  sm 
  \lr, 
  , 
  

  

  i 
  1 
  21 
  v 
  

  

  , 
  . 
  nl 
  S» 
  (ix.) 
  

  

  vA* 
  sin 
  — 
  , 
  

  

  . 
  - 
  - 
  v 
  vk 
  . 
  

  

  sin 
  a 
  sinn 
  p= 
  — 
  — 
  m 
  = 
  — 
  m 
  sin 
  y\ 
  

  

  With 
  these 
  substitutions, 
  (v.) 
  will 
  give 
  

  

  y 
  r+ 
  i 
  — 
  2y 
  r 
  cos6+y 
  r 
  - 
  x 
  = 
  Q 
  .... 
  (x.) 
  

  

  The 
  general 
  solution 
  of 
  this 
  set 
  of 
  equations 
  is 
  

  

  y 
  r 
  = 
  (Ce 
  ir9 
  + 
  De~ 
  ire 
  ) 
  e 
  int 
  , 
  

   = 
  f 
  Q>e 
  w< 
  *-~' 
  r 
  $ 
  + 
  D# 
  ~ 
  * 
  ra 
  + 
  rj8 
  ) 
  e 
  int 
  . 
  

  

  Now 
  (ix.) 
  will 
  not 
  determine 
  the 
  sign 
  of 
  yS 
  ; 
  we 
  will 
  always 
  

   take 
  the 
  positive 
  value. 
  It 
  is 
  then 
  obvious 
  that 
  D 
  = 
  ; 
  other- 
  

   wise 
  the 
  motion 
  would 
  be 
  great 
  for 
  great 
  values 
  of 
  r. 
  

  

  .*. 
  y 
  r 
  = 
  C^* 
  +n 
  '> 
  (xi.) 
  

  

  Again, 
  (vii.) 
  becomes 
  

  

  B-A 
  = 
  

  

  , 
  — 
  int 
  

  

  i 
  siw\fr 
  

   iC 
  

  

  0/i 
  +yo 
  c 
  os 
  ^— 
  2y 
  cos 
  6) 
  

  

  sin-\/r 
  

   Furthermore, 
  at 
  # 
  = 
  0, 
  

  

  From 
  (xii.) 
  and 
  (xiii.), 
  

  

  (e~ 
  ie 
  — 
  cos 
  yjr) 
  (xii.) 
  

  

  B 
  + 
  A 
  = 
  C 
  (xiii.) 
  

  

  2A 
  = 
  

  

  {-e-u 
  + 
  e- 
  1 
  *), 
  

   siny 
  ' 
  J 
  

  

  iG 
  

  

  sin-^r 
  

  

  iG 
  

  

  y 
  • 
  • 
  (xiv.) 
  

  

  