﻿504 
  Mr. 
  R. 
  V. 
  Southwell 
  on 
  the 
  

  

  considerable 
  length 
  will 
  collapse 
  into 
  a 
  two-lobed 
  form, 
  and 
  

   at 
  sensibly 
  the 
  same 
  pressure, 
  of 
  amount 
  

  

  » 
  c 
  =2-m 
  E 
  ' 
  ..... 
  (3) 
  

  

  This 
  is 
  the 
  theoretical 
  formula 
  for 
  the 
  collapsing 
  pres- 
  

   sure 
  of 
  a 
  tube 
  of 
  infinite 
  length, 
  first 
  given 
  by 
  Professor 
  

   Bryan 
  *. 
  

  

  Except 
  as 
  regards 
  the 
  numerical 
  value 
  of 
  the 
  collapsing 
  

   pressure, 
  these 
  conclusions 
  are 
  well 
  supported 
  by 
  the 
  results 
  

   of 
  experiment 
  f, 
  which 
  show 
  that 
  after 
  a 
  certain 
  limit,, 
  

   generally 
  termed 
  " 
  the 
  critical 
  length," 
  has 
  been 
  reached,, 
  

   no 
  appreciable 
  change 
  in 
  the 
  strength 
  of 
  a 
  tube 
  results 
  from 
  

   a 
  further 
  increase 
  of 
  its 
  length. 
  Hence, 
  the 
  formula 
  (3) 
  

   may 
  fairly 
  be 
  compared 
  with 
  the 
  results 
  of 
  experiments 
  on 
  

   tubes 
  of 
  finite 
  length, 
  provided 
  only 
  that 
  this 
  " 
  critical 
  

   length 
  " 
  is 
  exceeded. 
  

  

  The 
  experiments 
  of 
  Carman 
  and 
  Stewart 
  have 
  provided 
  

   full 
  and 
  accurate 
  data 
  for 
  the 
  comparison, 
  and 
  they 
  leave 
  

   no 
  room 
  for 
  doubt 
  that 
  tubes 
  of 
  practical 
  dimensions 
  and 
  

   material 
  will 
  collapse 
  under 
  pressures 
  very 
  much 
  less 
  than 
  

   the 
  formula 
  (3) 
  suggests. 
  This, 
  however, 
  is 
  not 
  surprising 
  

   if 
  we 
  remember 
  that 
  the 
  formula 
  is 
  based 
  upon 
  the 
  assumption 
  

   that 
  the 
  elasticity 
  of 
  the 
  material 
  is 
  perfect 
  at 
  the 
  instant 
  of 
  

   collapse, 
  whereas 
  the 
  pressure 
  given 
  by 
  (3) 
  will 
  be 
  more 
  than 
  

   sufficient 
  to 
  produce 
  elastic 
  break-down 
  in 
  tubes 
  of 
  ordinary 
  

   material, 
  unless 
  these 
  are 
  very 
  thin. 
  

  

  If 
  this 
  is 
  a 
  correct 
  explanation 
  of 
  the 
  discrepancy 
  between 
  

   theory 
  and 
  experiment, 
  we 
  shall 
  expect 
  to 
  find 
  that 
  the 
  

   formula 
  (3) 
  is 
  supported 
  best 
  by 
  experiments 
  on 
  very 
  thin 
  

   tubes, 
  and 
  this 
  is 
  actually 
  the 
  case. 
  On 
  the 
  other 
  hand 
  y 
  

   if 
  practical 
  imperfections 
  of 
  form 
  and 
  material 
  were 
  the 
  

   chief 
  causes 
  of 
  weakness, 
  as 
  has 
  sometimes 
  been 
  suggested, 
  

   it 
  would 
  be 
  reasonable 
  to 
  look 
  for 
  closest 
  agreement 
  to 
  

   experiments 
  on 
  thick 
  tubes, 
  in 
  which 
  greater 
  accuracy 
  of 
  

   workmanship 
  may 
  be 
  expected. 
  Similar 
  considerations 
  

   apply 
  in 
  the 
  case 
  of 
  the 
  strut 
  problem. 
  Euler's 
  formula 
  

   gives 
  its 
  closest 
  predictions 
  of 
  strength 
  in 
  the 
  case 
  of 
  

   long 
  struts, 
  which 
  are 
  comparatively 
  difficult 
  to 
  load 
  with 
  

   accuracy, 
  and 
  this 
  fact 
  makes 
  it 
  unlikely 
  that 
  its 
  untrust- 
  

   worthiness 
  in 
  the 
  case 
  of 
  shorter 
  struts 
  is 
  due 
  mainly 
  to 
  

   practical 
  imperfections. 
  

  

  * 
  Proc. 
  Camb. 
  Phil. 
  Soc. 
  vol. 
  vi. 
  p. 
  287 
  (1888). 
  

  

  t 
  Cf. 
  Carman 
  and 
  Stewart, 
  loc. 
  cit., 
  and 
  the 
  experimental 
  curve 
  given 
  

   in 
  my 
  second 
  paper 
  (footnote 
  j|, 
  p. 
  502). 
  

  

  