﻿APPENDIX 
  C, 
  347 
  

  

  " 
  Let 
  a 
  denote 
  the 
  distance 
  (in 
  feet) 
  of 
  this 
  point 
  or 
  edge 
  from 
  the 
  centre 
  

   of 
  gravity, 
  then 
  the 
  statical 
  work 
  done 
  in 
  upsetting 
  the 
  body, 
  whose 
  weight 
  

   is 
  W. 
  

  

  W 
  a 
  ( 
  i 
  — 
  cos. 
  <f). 
  

  

  "This 
  must 
  equal 
  the 
  dynamical 
  work 
  acquired, 
  which 
  (as 
  is 
  well 
  known) 
  is 
  

   equal 
  to 
  the 
  xvork 
  stored 
  up 
  in 
  the 
  centre 
  of 
  gyration, 
  or 
  

  

  W 
  a 
  ( 
  i 
  — 
  cos. 
  ) 
  = 
  ■ 
  

  

  Y 
  2g 
  

  

  where 
  w 
  is 
  the 
  angular 
  velocity 
  of 
  the 
  body 
  at 
  starting, 
  k 
  the 
  radius 
  of 
  

   gyration, 
  with 
  respect 
  to 
  the 
  point 
  or 
  arris 
  on 
  which 
  it 
  turns, 
  and 
  g 
  the 
  velo- 
  

   city 
  acquired 
  by 
  a 
  falling 
  body 
  in 
  one 
  second 
  of 
  time, 
  

   " 
  Equating 
  these 
  two 
  values 
  of 
  the 
  work 
  done 
  we 
  find— 
  

  

  u> 
  2 
  k 
  2 
  = 
  2 
  g 
  a 
  ( 
  i 
  — 
  cos. 
  <j>) 
  

  

  but 
  w, 
  the 
  angular 
  velocity, 
  is 
  equal 
  to 
  the 
  statical 
  couple 
  applied 
  divided 
  by 
  

   the 
  moment 
  of 
  inertia, 
  or 
  

  

  v 
  a 
  cos. 
  

  

  - 
  - 
  — 
  -p-* 
  

  

  squaring 
  and 
  substituting 
  

  

  „•= 
  2g 
  x 
  * 
  x 
  iSZ%L* 
  " 
  (I) 
  

  

  a 
  cos. 
  1 
  

  

  For 
  a 
  pillar 
  of 
  rectangular 
  section 
  

  

  *.* 
  + 
  * 
  

  

  and 
  a 
  = 
  i 
  y 
  a 
  + 
  /3- 
  

  

  where 
  a 
  is 
  the 
  height 
  and 
  j3 
  the 
  side 
  of 
  the 
  base 
  ; 
  hence 
  

  

  , 
  4 
  i 
  : 
  I 
  — 
  cos. 
  (t> 
  

  

  igX 
  y/^TW 
  X-^T^- 
  r 
  (I.) 
  

  

  p 
  = 
  !5^!_+i6a 
  2 
  

  

  For 
  a 
  cylindrical 
  column 
  

  

  V 
  a 
  2 
  + 
  f? 
  

  

  and 
  

   and 
  since 
  

  

  the 
  equation 
  (I) 
  becomes 
  

  

  12 
  V 
  X 
  £" 
  </ 
  a 
  a 
  + 
  /3 
  d 
  (i— 
  cos. 
  £t 
  ( 
  I 
  2) 
  

  

  ( 
  347 
  ) 
  

  

  