﻿48 
  Mr. 
  W. 
  H. 
  Macaulay 
  on 
  the 
  Stresses 
  

  

  that 
  is 
  to 
  say, 
  u 
  + 
  v=— 
  2 
  — 
  /3, 
  and 
  uv=l. 
  It 
  is 
  clear 
  that 
  

   u 
  and 
  v 
  are 
  both 
  negative 
  ; 
  let 
  us 
  denote 
  by 
  u 
  the 
  root 
  which 
  

   is 
  a 
  proper 
  fraction. 
  

  

  Similarly 
  R 
  m 
  , 
  R 
  m 
  +i, 
  . 
  . 
  . 
  Rn-i 
  are 
  given 
  by 
  the 
  equation 
  

  

  R 
  p 
  =AV- 
  w 
  + 
  B^- 
  w 
  , 
  . 
  . 
  . 
  . 
  (10) 
  

  

  and 
  

  

  aR 
  N 
  = 
  AV 
  l 
  + 
  BV 
  l 
  , 
  ... 
  . 
  . 
  . 
  (11) 
  

  

  where 
  

  

  A' 
  + 
  B' 
  = 
  A 
  + 
  B 
  (12) 
  

  

  We 
  have 
  now 
  used 
  N 
  — 
  3 
  of 
  our 
  N 
  equations 
  (1), 
  and 
  have 
  

   three 
  left 
  to 
  complete 
  the 
  determination 
  of 
  A 
  B 
  A' 
  B' 
  ? 
  namely 
  

   the 
  equation 
  

  

  R^i 
  + 
  (2+/9)R 
  m 
  + 
  R 
  w+1 
  -i/9W 
  = 
  0, 
  . 
  . 
  (13) 
  

   and 
  the 
  first 
  two 
  of 
  the 
  equations 
  (1), 
  which 
  are 
  

  

  Ri+IRi-R^Rs- 
  . 
  . 
  .-(-l)*R*»(a+/3) 
  + 
  ^W2^=0, 
  (1.4) 
  

  

  -R 
  1 
  (a 
  + 
  /3)+R 
  2 
  +{R 
  2 
  -R 
  3 
  + 
  . 
  . 
  . 
  + 
  (-l) 
  N 
  R 
  N 
  }(a 
  + 
  2/3) 
  

  

  + 
  -WV 
  2 
  =0 
  . 
  (15) 
  

  

  Now 
  from 
  equation 
  (2) 
  

  

  T 
  «= 
  i 
  > 
  W 
  + 
  R 
  1 
  -R 
  3 
  + 
  . 
  . 
  . 
  -(-1) 
  N 
  R 
  N 
  . 
  . 
  . 
  (16) 
  

   So 
  equations 
  (14) 
  and 
  (15) 
  may 
  be 
  written 
  

  

  R 
  1 
  + 
  T 
  (a 
  + 
  £)+W{-/> 
  (a 
  + 
  £) 
  + 
  J 
  2^ 
  1 
  } 
  = 
  0, 
  

  

  R 
  1 
  (3 
  + 
  ~R 
  2 
  -T 
  {a 
  + 
  2f3) 
  + 
  W{p 
  {a 
  + 
  2f3) 
  + 
  ^ 
  2«pM 
  = 
  0, 
  

   or 
  

  

  R 
  1 
  4-T 
  (a 
  + 
  /3)=-iW^ft 
  . 
  . 
  . 
  (17) 
  

  

  N 
  

  

  n 
  

  

  R 
  1 
  /3 
  + 
  R 
  2 
  -T 
  (a 
  + 
  2/3)=W^/3. 
  . 
  . 
  (18) 
  

  

  We 
  have 
  now 
  to 
  solve 
  equations 
  (12), 
  (13), 
  (16), 
  (17), 
  and 
  

   (18) 
  for 
  A, 
  B, 
  A', 
  B'. 
  

  

  Since 
  /3= 
  —(l 
  + 
  w) 
  2 
  / 
  w 
  > 
  equation 
  (13) 
  gives 
  

  

  &'u 
  + 
  B'v=Av 
  + 
  Bu- 
  { 
  ±p^W. 
  

  

  