﻿52 
  Mr. 
  W. 
  H. 
  Macaulay 
  on 
  the 
  Stresses 
  

  

  T 
  6p 
  =(p-i) 
  £ 
  *W 
  + 
  j^- 
  u 
  (Al^nH-l 
  + 
  Bup 
  -), 
  

  

  T 
  cp 
  = 
  - 
  i 
  J 
  sW 
  - 
  j^ 
  (A 
  M 
  — 
  m* 
  + 
  Biip-*) 
  , 
  

  

  T* 
  = 
  i 
  £ 
  *W 
  - 
  ^ 
  (A^-p+i 
  + 
  Bur-), 
  

  

  R„ 
  =Aw 
  m 
  "P 
  + 
  Bi« 
  - 
  m 
  (this 
  being 
  «T 
  when 
  p=0) 
  ; 
  

  

  T„ 
  p 
  = 
  -(N-/3 
  + 
  1) 
  ^W 
  + 
  ji- 
  (Ati^nP+i 
  + 
  BH 
  

  

  — 
  liW 
  z 
  (u 
  m 
  -p 
  +1 
  - 
  up~ 
  m 
  ), 
  

  

  T 
  5p 
  = 
  (N 
  - 
  P 
  + 
  i) 
  ~ 
  W 
  + 
  -— 
  ( 
  Att 
  w 
  -p 
  +1 
  + 
  Bmp-») 
  

  

  JN 
  1 
  -\-1L 
  

  

  —UW 
  — 
  — 
  (u 
  m 
  -p 
  +l 
  — 
  i(p- 
  m 
  ) 
  , 
  

   T 
  cp 
  =i^sW- 
  "— 
  (At^-^ 
  + 
  BuP-") 
  

  

  JN 
  1+ 
  W 
  

  

  t 
  ^p= 
  -i 
  S*W- 
  r^— 
  (A 
  w 
  «-p-+i 
  + 
  Bup-*») 
  

  

  JN 
  1 
  + 
  u 
  v 
  

  

  K 
  p 
  = 
  Au 
  m 
  ~ 
  p 
  + 
  Bz^p- 
  m 
  - 
  JW 
  y^ 
  (w 
  w 
  ~p 
  — 
  ?^" 
  m 
  ) 
  

   (this 
  being 
  «R 
  N 
  when 
  p 
  = 
  N). 
  

  

  In 
  these 
  expressions 
  the 
  values 
  which 
  have 
  been 
  found 
  for 
  

   A 
  and 
  B 
  can 
  be 
  inserted. 
  We 
  get 
  at 
  once 
  

  

  (A#-p 
  +1 
  + 
  B^p- 
  m 
  ) 
  =iW 
  

  

  1+M 
  V 
  J 
  ~ 
  2 
  $-^U 
  N 
  + 
  1 
  

  

  C(l—u 
  n 
  )(^ 
  + 
  ylru 
  n+1 
  X(j)U-P 
  — 
  ylruP)u 
  m+1 
  

  

  + 
  ^(^-p+i- 
  w 
  p)|. 
  . 
  (27) 
  

   The 
  full 
  expression 
  for 
  R 
  p 
  has 
  already 
  been 
  written 
  down. 
  

  

  