﻿62 
  Mr. 
  W. 
  H. 
  Macaulay 
  on 
  the 
  Stresses 
  

  

  and 
  

  

  ^o=-i(-l) 
  w 
  +^{m+(-l) 
  N 
  n}, 
  

  

  also 
  

  

  e 
  + 
  ter*= 
  2{v 
  + 
  * 
  2 
  2\+ 
  s 
  2 
  S 
  fi) 
  . 
  

  

  Thus 
  the 
  expression 
  for 
  R 
  is 
  very 
  simple, 
  namely, 
  

  

  m 
  my 
  

  

  If 
  N 
  is 
  even 
  and 
  m—n, 
  

  

  m 
  m 
  

  

  1 
  1 
  

  

  and 
  

  

  R=-i{l-(l) 
  n 
  }W, 
  

   which 
  verifies 
  the 
  remark 
  made 
  above. 
  

  

  For 
  the 
  case 
  of 
  a 
  uniform 
  loading 
  W 
  on 
  each 
  of 
  the 
  N— 
  1 
  

   lower 
  joints, 
  

  

  R(e 
  + 
  ter 
  2 
  ) 
  + 
  Wtterp 
  = 
  0. 
  

  

  Here 
  22<?rp 
  means 
  the 
  summation 
  of 
  all 
  values 
  of 
  %erp 
  for 
  

   different 
  values 
  of 
  m 
  and 
  n. 
  To 
  evaluate 
  it 
  let 
  us 
  find 
  first 
  

   the 
  part 
  of 
  it 
  which 
  belongs 
  to 
  the 
  two 
  bars 
  aa 
  and 
  ha. 
  We 
  

   have 
  

  

  p 
  a 
  ,+p^(-^-{-^+ 
  my 
  (<r=ito 
  m 
  ), 
  

  

  JVt^-Hr^l' 
  (<r=m 
  + 
  ltoN), 
  

  

  the 
  term 
  in 
  square 
  brackets 
  to 
  be 
  kept 
  only 
  if 
  n 
  is 
  odd. 
  

   Thus 
  the 
  part 
  of 
  %%erp 
  for 
  the 
  bars 
  aa 
  and 
  ba 
  is 
  

  

  M^{^--^^^> 
  + 
  (N-.)N 
  +[N 
  ] 
  }, 
  

  

  the 
  term 
  in 
  square 
  brackets 
  here 
  to 
  be 
  kept 
  only 
  if 
  N— 
  <r 
  is 
  

   odd 
  ; 
  this 
  reduces 
  to 
  

  

  ^^{N-C-lf-'f. 
  

  

  Similarly 
  the 
  part 
  of 
  %%erp 
  belonging 
  to 
  the 
  bars 
  ca 
  and 
  

   da 
  is 
  

  

  