﻿1004 
  Lord 
  Ilayleigh 
  on 
  the 
  Stability 
  of 
  the 
  

  

  value, 
  dvjdy 
  may 
  there 
  be 
  either 
  continuous 
  or 
  discon- 
  

   tinuous. 
  Even 
  when 
  there 
  is 
  but 
  one 
  place 
  where 
  ?i 
  + 
  HJ 
  = 
  

   with 
  the 
  proposed 
  value 
  o£ 
  n, 
  it 
  may 
  happen 
  that 
  dvjdy 
  is 
  

   there 
  continuous. 
  

  

  The 
  argument 
  above 
  employed 
  is 
  not 
  interfered 
  with 
  

   even 
  though 
  U 
  is 
  such 
  that 
  dTJ/dy 
  is 
  here 
  and 
  there 
  dis- 
  

   continuous, 
  so 
  as 
  to 
  make 
  d 
  2 
  \]/dy 
  2 
  infinite. 
  At 
  any 
  such 
  

   place 
  the 
  necessary 
  condition 
  is 
  obtained 
  by 
  integrating 
  (3) 
  

   across 
  the 
  discontinuity. 
  As 
  was 
  shown 
  in 
  my 
  former 
  

   paper 
  (loc. 
  cit;), 
  it 
  is 
  

  

  (;+4*(!)-*(f)-= 
  o.- 
  • 
  • 
  < 
  6 
  > 
  

  

  A 
  being 
  the 
  symbol 
  of 
  finite 
  differences 
  ; 
  and 
  by 
  (6) 
  the 
  

   corresponding 
  sudden 
  change 
  in 
  dvjdy 
  is 
  determined. 
  

  

  It 
  appears 
  then 
  that 
  any 
  value 
  of 
  —kU 
  is 
  a 
  possible 
  

   value 
  of 
  n. 
  Are 
  other 
  real 
  values 
  admissible 
  ? 
  If 
  so, 
  

   n-j-k\J 
  is 
  of 
  one 
  sign 
  throughout. 
  It 
  is 
  easy 
  to 
  see 
  that 
  if 
  

   d 
  2 
  JJ/dy 
  2 
  has 
  throughout 
  the 
  same 
  sign 
  as 
  n 
  + 
  kTJ, 
  no 
  solution 
  

   is 
  possible. 
  I 
  propose 
  to 
  prove 
  that 
  no 
  solution 
  is 
  possible 
  

   in 
  any 
  case 
  if 
  n 
  + 
  IcU, 
  being 
  real, 
  is 
  of 
  one 
  sign 
  throughout. 
  

  

  If 
  U' 
  be 
  written 
  for 
  U-f-??//f, 
  our 
  equation 
  (3) 
  takes 
  the 
  

   form 
  

  

  lv 
  $-^ 
  = 
  m 
  >- 
  ■ 
  ■ 
  • 
  (?) 
  

  

  or 
  on 
  integration 
  with 
  respect 
  to 
  y, 
  

  

  U^-»^=K 
  + 
  pf 
  y 
  U't)fe 
  ... 
  (8) 
  

  

  dy 
  dy 
  Jo 
  

  

  where 
  K 
  is 
  an 
  arbitrary 
  constant. 
  Assume 
  » 
  = 
  UV 
  ; 
  then 
  

  

  dv 
  l 
  K 
  P 
  Cy 
  ,-'„ 
  -. 
  

  

  Ty 
  = 
  W^w)/ 
  Wd 
  ^' 
  ' 
  * 
  ' 
  (9) 
  

  

  whence, 
  on 
  integration 
  and 
  replacement 
  of 
  v, 
  

  

  v 
  = 
  HU' 
  + 
  KU' 
  j^ 
  + 
  iW 
  (J^j/UWy, 
  . 
  (10) 
  

  

  H 
  denoting 
  a 
  second 
  arbitrary 
  constant. 
  

  

  In 
  (10) 
  we 
  may 
  suppose 
  y 
  measured 
  from 
  the 
  first 
  wall, 
  

   where 
  v=0. 
  Hence, 
  unless 
  U' 
  vanish 
  with 
  y, 
  H 
  = 
  0. 
  Also 
  

   from 
  (8) 
  when 
  y 
  = 
  0, 
  

  

  «).=* 
  • 
  (u) 
  

  

  