﻿54 
  Mr. 
  W. 
  H. 
  Macaulay 
  on 
  the 
  Stresses 
  

  

  Thus 
  Ri 
  %eqr± 
  + 
  'R 
  2 
  %eqr 
  2 
  + 
  . 
  . 
  . 
  

  

  =2</(Ri-R 
  9 
  + 
  ...)-(-l) 
  N 
  ^(Ri-2R 
  2 
  + 
  3R 
  3 
  -...)(^ 
  2 
  + 
  ^ 
  2 
  ) 
  

   + 
  {-l) 
  k 
  {K 
  1 
  -2R 
  2 
  -... 
  + 
  (-l) 
  k+1 
  kR 
  k 
  \{\t?+^ 
  

  

  + 
  A{R 
  t+1 
  -.R 
  t+2 
  + 
  . 
  . 
  . 
  +(-l) 
  i+1 
  R 
  N 
  }(^ 
  2 
  +^ 
  2 
  )- 
  I 
  

  

  Now 
  

  

  R,-R 
  2 
  + 
  ... 
  - 
  -(Aw™-f 
  Bt>™)-p 
  W, 
  

  

  and 
  the 
  other 
  sums 
  can 
  be 
  evaluated 
  in 
  terms 
  of 
  A, 
  B, 
  u, 
  v, 
  

   and, 
  finally, 
  we 
  get 
  for 
  the 
  value 
  of 
  the 
  whole 
  expression 
  

  

  \{- 
  2«i> 
  ?o"W 
  - 
  J* 
  (Au 
  m 
  + 
  Bv 
  m 
  ) 
  - 
  ~ 
  {A!u 
  n 
  + 
  B 
  V) 
  + 
  Au 
  m 
  ~ 
  k 
  + 
  Bv 
  m 
  ~ 
  k 
  1 
  

  

  -i(\i 
  2 
  +^ 
  2 
  )W^{[l-(-l)' 
  l 
  -(-l) 
  / 
  ]m+(-l)" 
  +w 
  N}. 
  

   Thus 
  

  

  + 
  *W^ 
  2 
  ^-^ 
  [^(CT 
  + 
  *Rir)-B*}v 
  • 
  (29) 
  

  

  This 
  is 
  the 
  deflection 
  at 
  the 
  point 
  k, 
  I 
  when 
  k<m-\-l, 
  and 
  by 
  

   symmetry 
  we 
  see 
  that 
  the 
  deflection 
  at 
  this 
  point 
  when 
  k>m 
  is 
  

  

  $W\t^ 
  (N 
  2 
  -m*-?-i) 
  +iW^ 
  

  

  -|{^(2T 
  +^E 
  N 
  )-^|. 
  . 
  (30) 
  

  

  If 
  we 
  substitute 
  for 
  T 
  , 
  Rh, 
  and 
  Bj 
  the 
  values 
  which 
  we 
  have 
  

   found 
  for 
  them 
  (25), 
  we 
  get 
  for 
  the 
  deflection 
  when 
  k< 
  m 
  + 
  1, 
  

  

  1 
  + 
  u 
  f 
  (1 
  — 
  u 
  n 
  )(l-u 
  k 
  )((j) 
  + 
  ylru 
  n 
  + 
  1 
  )(^u 
  m 
  - 
  k 
  j-yfru 
  m+1 
  ) 
  

  

  + 
  jp(«iA 
  + 
  Zn) 
  ft-uT) 
  + 
  | 
  (^-<)- 
  ^O*-^) 
  } 
  .; 
  .(31) 
  

   and 
  when 
  k 
  > 
  m, 
  

  

  