﻿.46 
  Mr. 
  W. 
  H. 
  Macaulny 
  on 
  the 
  Stresses 
  

  

  all 
  the 
  different 
  cases 
  which 
  arise 
  according 
  as 
  m 
  and 
  n 
  are 
  

   odd 
  or 
  even. 
  

  

  Weget*>=(-l)»+\ 
  

  

  if 
  cr<p 
  + 
  l, 
  r 
  Pa<r 
  =r 
  pW 
  —{—l)P 
  + 
  % 
  

  

  r 
  P 
  c<r=r 
  pd(r 
  ={-iy 
  + 
  ° 
  +1 
  s, 
  

  

  thus 
  ( 
  - 
  1) 
  P 
  + 
  *ter 
  p 
  r 
  a 
  = 
  j/ 
  + 
  2p(\t 
  2 
  + 
  fxs 
  2 
  ) 
  , 
  

  

  where 
  p<o- 
  + 
  l, 
  

  

  the 
  summation 
  being 
  for 
  all 
  bars 
  of 
  the 
  just 
  stiff 
  frame, 
  r 
  p 
  

   referring 
  to 
  the 
  pth, 
  and 
  r„ 
  to 
  the 
  o-th 
  redundant 
  bar. 
  

  

  For 
  the 
  panels 
  1 
  to 
  m 
  : 
  

  

  if 
  N 
  — 
  <r 
  is 
  eYen,p 
  a(X 
  = 
  -o-^t+\t], 
  p 
  h(T 
  = 
  (a 
  — 
  1)^ 
  t-\- 
  \t\, 
  

  

  Pea 
  = 
  — 
  H, 
  Prfcr 
  = 
  j$S- 
  [s] 
  ; 
  

  

  if 
  N-o-is 
  odd, 
  p 
  aa 
  = 
  -(<7— 
  1)^« 
  — 
  [*], 
  ?i<r=ff^-Wj 
  

  

  where 
  the 
  terms 
  in 
  square 
  brackets 
  are 
  to 
  be 
  kept 
  if 
  n 
  is 
  

   odd, 
  and 
  to 
  be 
  omitted 
  if 
  n 
  is 
  even. 
  

   For 
  the 
  panels 
  m 
  + 
  1 
  to 
  N 
  : 
  

  

  if 
  N-<riseven,-/w 
  = 
  -(N-o-)^V 
  /> 
  6<J 
  . 
  = 
  (N-<r 
  + 
  l)^«, 
  

  

  m 
  

  

  JOexr 
  =0, 
  Pd<r 
  = 
  — 
  J^ 
  S 
  ', 
  

  

  if 
  N-o- 
  is 
  odd, 
  p 
  a 
  , 
  = 
  -(N- 
  cr+ 
  1) 
  ^*, 
  Ffe 
  =(N- 
  o") 
  ^,- 
  

  

  Thus 
  for 
  the 
  panels 
  1 
  to 
  m, 
  

  

  ^+^ 
  = 
  (-l) 
  N 
  -'{^-[2«]}; 
  

  

  