﻿488 
  Mr. 
  W. 
  J. 
  Harrison 
  on 
  the 
  

  

  The 
  conditions 
  are 
  

  

  (1) 
  H=r 
  = 
  zy 
  = 
  (tj=—Ji), 
  

  

  (2) 
  u- 
  = 
  (.- 
  = 
  0, 
  c 
  = 
  6), 
  

  

  Some 
  o£ 
  these 
  conditions 
  are 
  satisfied 
  identically 
  ; 
  from 
  the 
  

   remainder 
  we 
  derive 
  the 
  six 
  equations 
  

  

  a(a 
  + 
  2v 
  m-) 
  A^ 
  + 
  gm 
  A3 
  + 
  (/ 
  U 
  + 
  2vm' 
  a 
  Y 
  = 
  0, 
  (1) 
  

  

  2i 
  hn 
  A2 
  + 
  ih 
  U 
  + 
  m^ 
  ^Y 
  -= 
  0, 
  (2) 
  

  

  2mn 
  7r/hA2-rn 
  7r/6 
  U 
  + 
  m' 
  X 
  = 
  0, 
  (3) 
  

  

  — 
  m 
  Ai 
  sink 
  mJi 
  + 
  m 
  A2 
  cosh 
  mh 
  + 
  U 
  cosh 
  m^Ji—Y 
  sinh 
  ??i7i 
  = 
  0, 
  (4) 
  

  

  ik 
  A 
  I 
  cosh 
  mil 
  — 
  ik 
  A2 
  sinh 
  mh 
  — 
  W 
  sinh 
  7nVi 
  + 
  Y 
  cosh 
  nih 
  = 
  0^ 
  (5) 
  

  

  nir/b 
  Ai 
  cosh 
  nih 
  + 
  nir/b 
  A2 
  sinh 
  mh 
  + 
  X 
  sinh 
  /;i'/i 
  — 
  Z 
  cosh 
  ju'A 
  = 
  0, 
  (6) 
  

  

  ^yhere 
  

  

  y 
  = 
  ;;7r/6F2-z7jHi, 
  

  

  W 
  = 
  ?H'Hi-717r/^.G2, 
  

  

  X 
  = 
  m'F2-zl'G2, 
  

   Y 
  = 
  m' 
  Ho 
  — 
  nTT/'h 
  Gi, 
  

   Z 
  = 
  m'F2-z7G,. 
  

   Among 
  these 
  six 
  quantities 
  there 
  exist 
  the 
  two 
  relations 
  

  

  m'\] 
  + 
  il^V-n7T/hX 
  = 
  0, 
  (7) 
  

   m'Y 
  + 
  ikY-n7r/bZ=0. 
  (8) 
  

  

  From 
  equations 
  (1) 
  to 
  (8) 
  we 
  eliminate 
  Ai, 
  A2, 
  U, 
  V, 
  W, 
  

   X, 
  Yj'Z, 
  and 
  obtain 
  the 
  period 
  equation. 
  

  

  Approximating 
  on 
  the 
  assumption 
  that 
  m'h 
  is 
  large 
  by 
  

   reason 
  of 
  the 
  smallness 
  of 
  v, 
  we 
  find 
  that, 
  if 
  we 
  replace 
  k 
  by 
  

   in 
  in 
  the 
  results 
  for 
  waves 
  propagated 
  in 
  one 
  direction, 
  we 
  

   obtain 
  the 
  corresponding 
  solution 
  for 
  this 
  problem. 
  This 
  

  

  