﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  HI 
  

  

  By 
  eliminating 
  — 
  between 
  (54.) 
  and 
  (55.) 
  we 
  have 
  

  

  86 
  n 
  „ 
  r 
  6 
  /SO 
  

  

  r 
  2 
  6 
  6 
  r 
  6 
  /6 
  6\ 
  3 
  

  

  When 
  P=0, 
  M 
  depends 
  on 
  the 
  sixth 
  power 
  of 
  the 
  radius 
  and 
  the 
  cube 
  of 
  the 
  

   angle 
  of 
  torsion, 
  when 
  the 
  cylinder 
  is 
  composed 
  of 
  separate 
  filaments. 
  

  

  Since 
  the 
  force 
  of 
  torsion 
  for 
  a 
  homogeneous 
  cylinder 
  depends 
  on 
  the 
  fourth 
  

   power 
  of 
  the 
  radius 
  and 
  the 
  first 
  power 
  of 
  the 
  angle 
  of 
  torsion, 
  the 
  torsion 
  of 
  a 
  

   wire 
  having 
  a 
  fibrous 
  texture 
  will 
  depend 
  on 
  both 
  these 
  laws. 
  

  

  The 
  parts 
  of 
  the 
  force 
  of 
  torsion 
  which 
  depend 
  on 
  these 
  two 
  laws 
  may 
  be 
  

   found 
  by 
  experiment, 
  and 
  thus 
  the 
  difference 
  of 
  the 
  elasticities 
  in 
  the 
  direction 
  of 
  

   the 
  axis 
  and 
  in 
  the 
  perpendicular 
  directions 
  may 
  be 
  determined. 
  

  

  A 
  calculation 
  of 
  the 
  force 
  of 
  torsion, 
  on 
  this 
  supposition, 
  may 
  be 
  found 
  in 
  

   Young's 
  Mathematical 
  Principles 
  of 
  Natural 
  Philosophy 
  ; 
  and 
  it 
  is 
  introduced 
  

   here 
  to 
  account 
  for 
  the 
  variations 
  from 
  the 
  law 
  of 
  Case 
  II., 
  which 
  may 
  be 
  observed 
  

   in 
  a 
  twisted 
  rod. 
  

  

  Case 
  VIII. 
  

  

  It 
  is 
  well 
  known 
  that 
  grindstones 
  and 
  fly-wheels 
  are 
  often 
  broken 
  by 
  the 
  

   centrifugal 
  force 
  produced 
  by 
  their 
  rapid 
  rotation. 
  I 
  have 
  therefore 
  calculated 
  

   the 
  strains 
  and 
  pressure 
  acting 
  on 
  an 
  elastic 
  cylinder 
  revolving 
  round 
  its 
  axis, 
  and 
  

   acted 
  on 
  by 
  the 
  centrifugal 
  force 
  alone. 
  

  

  The 
  equation 
  of 
  the 
  equilibrium 
  of 
  a 
  particle 
  (see 
  Equation 
  (21.)), 
  becomes 
  

  

  dp 
  4tt 
  2 
  A; 
  „ 
  

   * 
  y 
  dr 
  gt 
  2 
  

  

  where 
  q 
  and 
  p 
  are 
  the 
  tangential 
  and 
  radial 
  pressures, 
  k 
  is 
  the 
  weight 
  in 
  pounds 
  

   of 
  a 
  cubic 
  inch 
  of 
  the 
  substance, 
  g 
  is 
  twice 
  the 
  height 
  in 
  inches 
  that 
  a 
  body 
  falls 
  

   in 
  a 
  second, 
  t 
  is 
  the 
  time 
  of 
  revolution 
  of 
  the 
  cylinder 
  in 
  seconds. 
  

  

  By 
  substituting 
  the 
  value 
  of 
  g 
  and 
  ^| 
  in 
  Equations 
  (19.), 
  (20.), 
  and 
  neglect- 
  

   ing 
  o, 
  

  

  \9 
  fl 
  3mJ 
  \ 
  dr 
  g 
  t- 
  dr 
  2 
  ) 
  m 
  \ 
  dr 
  g 
  t 
  2 
  dr 
  2 
  J 
  

  

  which 
  gives 
  P 
  =c 
  ^ 
  + 
  2g~i 
  2 
  \ 
  2 
  + 
  m) 
  r+c 
  * 
  

  

  q-p=-c— 
  + 
  - 
  — 
  -(-4+ 
  — 
  )r 
  2 
  

   1 
  r 
  1 
  r 
  2 
  2 
  g 
  I 
  \ 
  m 
  ) 
  

  

  ... 
  (57.) 
  

  

  1 
  ?P±( 
  _ 
  9 
  3E 
  

  

  2gf\ 
  + 
  ^7' 
  ^ 
  2 
  / 
  

  

  If 
  the 
  radii 
  of 
  the 
  surfaces 
  of 
  the 
  hollow 
  cylinder 
  be 
  a 
  x 
  and 
  a 
  v 
  and 
  the 
  pres- 
  

   sures 
  acting 
  on 
  them 
  \ 
  and 
  k 
  2 
  , 
  then 
  the 
  values 
  of 
  c 
  x 
  and 
  c 
  2 
  are 
  

  

  VOL. 
  XX. 
  PART 
  I. 
  2 
  G 
  

  

  