﻿50 
  Mr. 
  W. 
  H. 
  Macaulay 
  on 
  the 
  Stresses 
  

  

  We 
  get 
  A' 
  and 
  B' 
  by 
  interchanging 
  m 
  and 
  n 
  in 
  these 
  

   results, 
  or 
  we 
  can 
  use 
  equations 
  (19). 
  

  

  Thus 
  the 
  solution 
  of 
  Maxwell's 
  equations 
  for 
  the 
  tensions 
  of 
  

   the 
  redundant 
  bars 
  of 
  our 
  frame 
  has 
  been 
  obtained 
  in 
  a 
  fairly 
  

   simple 
  and 
  concise 
  form. 
  

  

  R 
  p 
  (p 
  from 
  1 
  to 
  w) 
  = 
  Aw 
  m 
  -p 
  + 
  B?«p-" 
  l 
  ? 
  

   R 
  p 
  (/o 
  from 
  m 
  to 
  N 
  — 
  l)=A'uP- 
  m 
  + 
  B 
  f 
  u 
  m 
  -<>=Au 
  m 
  -< 
  i 
  + 
  ~BuP- 
  m 
  

  

  l—u 
  2 
  v 
  ' 
  

  

  Thus 
  the 
  general 
  expression 
  for 
  Ep 
  (p 
  from 
  1 
  to 
  N 
  •— 
  1) 
  may 
  

   be 
  written 
  

  

  2 
  (l-»)((/. 
  2 
  -\frV 
  N+2 
  ) 
  

  

  U 
  m 
  ~P 
  

  

  where 
  the 
  term 
  in 
  the 
  square 
  brackets 
  is 
  to 
  be 
  omitted 
  when 
  

   p— 
  m 
  is 
  negative. 
  

  

  This 
  expression 
  is 
  equal 
  to 
  «T 
  when 
  p 
  = 
  0, 
  and 
  to 
  «R 
  N 
  when 
  

   p 
  = 
  N. 
  It 
  should 
  be 
  noted 
  that 
  the 
  result 
  involves 
  the 
  ex- 
  

   tensibilities 
  of 
  the 
  bars 
  and 
  the 
  inclinations 
  of 
  the 
  sloping 
  

   ones 
  only 
  in 
  the 
  form 
  in 
  which 
  these 
  quantities 
  occur 
  in 
  a 
  

   and 
  ft. 
  When 
  v 
  l 
  = 
  v, 
  </> 
  and 
  yjr 
  are 
  each 
  unity. 
  Now 
  — 
  u 
  is 
  

   a 
  positive 
  proper 
  fraction 
  (except 
  in 
  the 
  limit 
  when 
  v 
  is 
  

   infinite); 
  and 
  it 
  is 
  clear 
  that 
  if 
  the 
  intermediate 
  vertical 
  

   bars 
  are 
  not 
  very 
  slender 
  compared 
  with 
  both 
  horizontal 
  

   and 
  diagonal 
  bars, 
  w 
  N 
  will 
  be 
  small, 
  and 
  good 
  approximate 
  

   results 
  of 
  a 
  simple 
  form 
  will 
  be 
  obtained 
  by 
  neglecting 
  

   this, 
  and 
  possibly 
  some 
  other 
  powers 
  of 
  u, 
  in 
  comparison 
  

   with 
  unity. 
  

  

  We 
  have 
  (1 
  -f 
  u)/a 
  = 
  <f> 
  + 
  it 
  ; 
  thus 
  

  

  1 
  r 
  (l-u»)(0 
  + 
  foi 
  « 
  +1 
  ) 
  _ 
  n 
  (l 
  _ 
  N) 
  ) 
  cj> 
  + 
  u 
  W 
  

   io 
  -\ 
  . 
  t 
  + 
  fu*^ 
  U 
  N 
  li 
  U 
  '/ 
  ^-^H-i-g 
  * 
  (2b) 
  

  

  If 
  we 
  put 
  v 
  = 
  co 
  in 
  this 
  equation 
  we 
  get 
  results 
  for 
  the 
  

  

  girder 
  /\/000\l 
  which 
  can 
  be 
  easily 
  verified. 
  

  

  The 
  remaining 
  tensions 
  may 
  be 
  found 
  either 
  from 
  Max- 
  

   well's 
  equation 
  (2) 
  or 
  from 
  the 
  equations 
  (4) 
  to 
  (7). 
  As 
  

   equations 
  (7) 
  are 
  not 
  very 
  convenient 
  in 
  form, 
  let 
  us 
  employ 
  

   equations 
  (2) 
  to 
  find 
  T„ 
  and 
  T 
  c 
  . 
  

  

  