﻿R 
  2 
  € 
  2 
  + 
  RiSeTV 
  2 
  + 
  'R^erf 
  + 
  R 
  3 
  2er 
  2 
  r 
  3 
  + 
  . 
  . 
  . 
  + 
  ~Fl<ep 
  

  

  3,} 
  <" 
  

  

  44 
  Mr. 
  W. 
  H. 
  Macaulay 
  on 
  the 
  Stresses 
  

  

  their 
  tensions 
  due 
  to 
  a 
  unit 
  tension 
  in 
  the 
  line 
  of 
  the 
  second 
  

   redundant 
  bar, 
  and 
  so 
  on 
  ; 
  then 
  R 
  1? 
  R 
  2 
  , 
  &c. 
  are 
  given 
  by 
  the 
  

   equations 
  

  

  R^x 
  + 
  R£ei\ 
  2 
  + 
  R 
  2 
  ^er 
  1 
  r 
  2 
  + 
  ^ter^ 
  + 
  • 
  • 
  • 
  + 
  FS^p^i 
  = 
  

   R 
  2 
  < 
  

  

  &c. 
  

  

  Also 
  Tj 
  =^ 
  + 
  ^1 
  + 
  ^2^2 
  + 
  (2) 
  

  

  &c. 
  

   and 
  

  

  «'*?=FS£/9<7 
  + 
  R 
  1 
  2£9Ti 
  + 
  R 
  2 
  X62t 
  2 
  + 
  (3) 
  

  

  To 
  apply 
  this 
  solution 
  to 
  the 
  case 
  of 
  a 
  girder 
  on 
  two 
  rigid 
  

   supports 
  with 
  a 
  load 
  at 
  one 
  of 
  the 
  joints 
  K, 
  we 
  may 
  suppose 
  

   that 
  a 
  base 
  connecting 
  the 
  supports 
  with 
  the 
  necessary 
  rigidity 
  

   forms 
  part 
  of 
  the 
  frame, 
  and 
  that 
  a 
  force 
  equal 
  to 
  the 
  load 
  

   acts 
  vertically 
  between 
  K 
  and 
  a 
  point 
  of 
  this 
  base 
  ; 
  the 
  formulge 
  

   given 
  by 
  Maxwell 
  then 
  apply, 
  and, 
  if 
  we 
  take 
  N 
  to 
  be 
  a 
  point 
  

   of 
  this 
  base 
  vertically 
  below 
  M, 
  the 
  value 
  we 
  get 
  for 
  x 
  will 
  be 
  

   the 
  deflection 
  of 
  the 
  joint 
  M. 
  If 
  there 
  are 
  several 
  loads, 
  we 
  

   must 
  investigate 
  their 
  effects 
  separately 
  and 
  compound 
  the 
  

   results. 
  

  

  Let 
  us 
  apply 
  these 
  equations 
  to 
  the 
  case 
  of 
  a 
  pin-jointed 
  

   girder 
  of 
  the 
  type 
  

  

  \/\/|\/ 
  

  

  A^ 
  

  

  /M/\ 
  

  

  \/ 
  v 
  

  

  . 
  

  

  v 
  

  

  < 
  

  

  with 
  N 
  panels, 
  with 
  all 
  the 
  horizontal 
  members 
  of 
  equal 
  ex- 
  

   tensibility 
  X, 
  all 
  the 
  sloping 
  members 
  of 
  equal 
  extensibility 
  /*, 
  

   all 
  the 
  N— 
  1 
  intermediate 
  vertical 
  members 
  of 
  equal 
  extensi- 
  

   bility 
  v, 
  and 
  the 
  two 
  end 
  vertical 
  members 
  of 
  equal 
  extensi- 
  

   bility 
  /. 
  Suppose 
  the 
  sloping 
  bars 
  not 
  to 
  be 
  pinned 
  together 
  

   at 
  their 
  intersections. 
  Let 
  6 
  be 
  the 
  inclination 
  to 
  the 
  vertical 
  

   of 
  each 
  of 
  the 
  sloping 
  bars, 
  and 
  let 
  us 
  write 
  s 
  and 
  t 
  for 
  sec 
  6 
  

   and 
  tan 
  6 
  respectively. 
  

  

  Let 
  the 
  ends 
  (lower 
  joints) 
  of 
  the 
  girders 
  rest 
  on 
  rigid 
  

   supports 
  A, 
  B, 
  and 
  a 
  load 
  W 
  be 
  applied 
  at 
  one 
  of 
  the 
  lower 
  

   joints 
  0, 
  and 
  let 
  the 
  weight 
  of 
  the 
  girder 
  be 
  neglected. 
  Let 
  

   m 
  be 
  the 
  number 
  of 
  panels 
  between 
  A 
  and 
  0, 
  and 
  n 
  the 
  

   number 
  between 
  and 
  B. 
  

  

  Distinguish 
  the 
  vertical 
  bar 
  at 
  A 
  by 
  the 
  suffix 
  0, 
  and 
  the 
  

  

  panels 
  from 
  A 
  to 
  B 
  each 
  consisting 
  of 
  the 
  bars 
  ">\] 
  by 
  the 
  

  

  suffixes 
  1, 
  2, 
  .... 
  N. 
  Select 
  for 
  a 
  just 
  stiff 
  frame 
  that 
  

   obtained 
  by 
  rejecting 
  all 
  the 
  vertical 
  bars 
  except 
  the 
  one 
  at 
  A, 
  

   then 
  the 
  tensions 
  of 
  these 
  vertical 
  bars 
  are 
  R 
  x 
  R 
  2 
  . 
  . 
  . 
  . 
  R 
  N 
  

  

  