﻿8 
  Lord 
  Rayleigh 
  on 
  the 
  

  

  If 
  we 
  introduce 
  the 
  value 
  o£ 
  /3, 
  the 
  equation 
  of 
  the 
  curve 
  

   may 
  be 
  written 
  

  

  h 
  = 
  k 
  x 
  ' 
  c 
  , 
  (28) 
  

  

  When 
  we 
  determine 
  k 
  so 
  as 
  to 
  make 
  P 
  a 
  maximum, 
  we 
  get 
  

   £ 
  = 
  2'3, 
  and 
  

  

  P 
  = 
  0-165^g- 
  2 
  , 
  (29) 
  

  

  again 
  with 
  an 
  advantage 
  which 
  is 
  but 
  small. 
  

  

  In 
  all 
  the 
  cases 
  so 
  far 
  considered 
  the 
  thickness 
  h 
  increases 
  

   all 
  the 
  way 
  along 
  the 
  length, 
  and 
  the 
  resultant 
  pressure 
  is 
  

   proportional 
  to 
  the 
  square 
  of 
  this 
  length 
  (c). 
  In 
  view 
  of 
  

   some 
  suggestions 
  which 
  have 
  been 
  made, 
  it 
  is 
  of 
  interest 
  to 
  

   inquire 
  what 
  is 
  the 
  effect 
  of 
  (say) 
  r 
  repetitions 
  of 
  the 
  same 
  

   curve, 
  as, 
  for 
  instance, 
  a 
  succession 
  ef 
  inclined 
  lines 
  

   ABCDEF 
  (fig. 
  1). 
  It 
  appears 
  from 
  (8) 
  that 
  H 
  has 
  the 
  

  

  Fig. 
  1. 
  

  

  U 
  

  

  same 
  value 
  for 
  the 
  aggregate 
  as 
  for 
  each 
  member 
  singly, 
  

   and 
  from 
  (5) 
  that 
  the 
  increment 
  of 
  p 
  in 
  passing 
  along 
  the 
  

   series 
  is 
  r 
  times 
  the 
  increment 
  due 
  to 
  one 
  member. 
  Since 
  

   the 
  former 
  increment 
  is 
  zero, 
  it 
  follows 
  that 
  the 
  pressure 
  is 
  

   zero 
  at 
  the 
  beginning 
  and 
  end 
  of 
  each 
  member. 
  The 
  

   circumstances 
  are 
  thus 
  precisely 
  the 
  same 
  for 
  each 
  member, 
  

   and 
  the 
  total 
  pressure 
  is 
  r 
  times 
  that 
  due 
  to 
  the 
  first, 
  sup- 
  

   posed 
  to 
  be 
  isolated. 
  But 
  if 
  we 
  imagine 
  the 
  curve 
  spread 
  

   once 
  over 
  the 
  entire 
  length 
  by 
  merely 
  increasing 
  the 
  scale 
  

   of 
  x, 
  we 
  see 
  that 
  the 
  resultant 
  pressure 
  would 
  be 
  increased 
  

   r 
  2 
  times, 
  instead 
  of 
  merely 
  r 
  times. 
  Accordingly 
  a 
  repeti- 
  

   tion 
  of 
  a 
  curve 
  is 
  very 
  unfavourable. 
  But 
  at 
  this 
  point 
  it 
  is 
  

   well 
  to 
  recall 
  that 
  we 
  are 
  limiting 
  ourselves 
  to 
  the 
  case 
  of 
  

   two 
  dimensions. 
  An 
  extension 
  in 
  the 
  third 
  dimension, 
  which 
  

   would 
  suffice 
  for 
  a 
  particular 
  length, 
  might 
  be 
  inadequate 
  

   when 
  this 
  length 
  is 
  multiplied 
  r 
  times. 
  

  

  The 
  forms 
  of 
  curve 
  hitherto 
  examined 
  have 
  been 
  chosen 
  

   with 
  regard 
  to 
  practical 
  or 
  mathematical 
  convenience, 
  and 
  

   it 
  remains 
  open 
  to 
  find 
  the 
  form 
  which 
  according 
  to 
  (5) 
  

   makes 
  P 
  a 
  maximum, 
  subject 
  to 
  the 
  conditions 
  of 
  a 
  given 
  

   length 
  and 
  a 
  given 
  minimum 
  thickness 
  (7^) 
  of 
  the 
  layer 
  of 
  

  

  