﻿96 
  MR 
  JAMES 
  CLERK 
  MAXWELL 
  ON 
  THE 
  

  

  In 
  these 
  cases 
  the 
  co-ordinates 
  x, 
  y, 
  z 
  are 
  replaced 
  by 
  the 
  co-ordinates 
  

   x=X, 
  measured 
  along 
  the 
  axis 
  of 
  the 
  cylinder. 
  

   y=r, 
  the 
  radius 
  of 
  any 
  point, 
  or 
  the 
  distance 
  from 
  the 
  axis. 
  

   z=rd, 
  the 
  arc 
  of 
  a 
  circle 
  measured 
  from 
  a 
  fixed 
  plane 
  passing 
  

   through 
  the 
  axis. 
  

  

  d 
  8 
  x 
  _d 
  Ox 
  

   dx 
  ~~ 
  dx 
  

  

  d 
  8 
  y 
  _d 
  8 
  r 
  

  

  dy 
  ~ 
  dr 
  

  

  j9 
  1= 
  0, 
  are 
  the 
  compression 
  and 
  pressure 
  in 
  the 
  direction 
  of 
  the 
  

   axis 
  at 
  any 
  point. 
  

  

  , 
  p 
  2 
  =p, 
  are 
  the 
  compression 
  and 
  pressure 
  in 
  the 
  direction 
  of 
  the 
  

   radius. 
  

  

  -t—=-j 
  — 
  -q= 
  — 
  . 
  p%=9-> 
  are 
  the 
  compression 
  and 
  pressure 
  in 
  the 
  direction 
  of 
  the 
  

  

  tangent. 
  

   Equations 
  (9.) 
  become, 
  when 
  expressed 
  in 
  terms 
  of 
  these 
  co-ordinates 
  — 
  

  

  m 
  

  

  d86 
  

  

  " 
  dr 
  

  

  m 
  

   9 
  2 
  = 
  ~2 
  

  

  d8e 
  

  

  dx 
  

  

  m 
  

   9 
  3 
  = 
  ^2 
  

  

  d 
  8 
  x 
  

   ' 
  dr 
  

  

  (14.) 
  

  

  ' 
  

  

  The 
  length 
  of 
  the 
  cylinder 
  is 
  b, 
  and 
  the 
  two 
  radii 
  a 
  x 
  and 
  a. 
  2 
  in 
  every 
  case. 
  

  

  Case 
  I. 
  

  

  The 
  first 
  equation 
  is 
  applicable 
  to 
  the 
  case 
  of 
  a 
  hollow 
  cylinder, 
  of 
  which 
  the 
  

   outer 
  surface 
  is 
  fixed, 
  while 
  the 
  inner 
  surface 
  is 
  made 
  to 
  turn 
  through 
  a 
  small 
  

   angle 
  8 
  6, 
  by 
  a 
  couple 
  whose 
  moment 
  is 
  M. 
  

  

  The 
  twisting 
  force 
  M 
  is 
  resisted 
  only 
  by 
  the 
  elasticity 
  of 
  the 
  solid, 
  and 
  there- 
  

   fore 
  the 
  whole 
  resistance, 
  in 
  every 
  concentric 
  cylindric 
  surface, 
  must 
  be 
  equal 
  to 
  M. 
  

  

  The 
  resistance 
  at 
  any 
  point, 
  multiplied 
  into 
  the 
  radius 
  at 
  which 
  it 
  acts, 
  is 
  ex- 
  

  

  m 
  

  

  pressed 
  by 
  rq 
  1 
  = 
  1 
  r-r 
  

  

  d86 
  

  

  2 
  dr 
  ' 
  

  

  Therefore 
  for 
  the 
  whole 
  cylindric 
  surface 
  

  

  d 
  8 
  6 
  r. 
  , 
  ,, 
  

  

  — 
  z— 
  m7T 
  r 
  6 
  o 
  = 
  M. 
  

   dr 
  

  

  Whence 
  88 
  = 
  = 
  1 
  ( 
  — 
  § 
  — 
  — 
  Y 
  ) 
  

  

  2 
  ir 
  m 
  o 
  \rtj 
  a 
  2 
  J 
  / 
  

  

  and 
  m 
  = 
  2-¥Y8-d(-aJ 
  ~ 
  ^) 
  ' 
  ' 
  (15 
  ° 
  

  

  The 
  optical 
  effect 
  of 
  the 
  pressure 
  of 
  any 
  point 
  is 
  expressed 
  by 
  

  

  t 
  a 
  Ub 
  nay 
  

  

  I 
  = 
  teg. 
  b 
  = 
  (a) 
  s- 
  . 
  . 
  (16.) 
  

  

  