﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  119 
  

  

  and 
  since 
  (g-p), 
  R 
  and 
  S 
  are 
  known, 
  and 
  since 
  at 
  the 
  surface, 
  where 
  (p 
  5 
  (z 
  1 
  y)=Q, 
  

   jo=0, 
  all 
  the 
  data 
  are 
  given 
  for 
  determining 
  the 
  absolute 
  value 
  of 
  p 
  by 
  integration. 
  

  

  Though 
  this 
  is 
  the 
  best 
  method 
  of 
  finding 
  p 
  and 
  q 
  by 
  graphic 
  construction, 
  it 
  

   is 
  much 
  better, 
  when 
  the 
  equations 
  of 
  the 
  curves 
  have 
  been 
  found, 
  that 
  is, 
  when 
  

   </> 
  1 
  and 
  <p 
  2 
  are 
  known, 
  to 
  resolve 
  the 
  pressures 
  in 
  the 
  direction 
  of 
  the 
  axes. 
  

  

  The 
  new 
  quantities 
  are 
  p 
  v 
  p 
  2 
  , 
  and 
  q 
  3 
  ; 
  and 
  the 
  equations 
  are 
  

  

  tan0= 
  — 
  &_, 
  (p-9) 
  2 
  =g 
  i 
  2 
  + 
  (p 
  1 
  -P 
  2 
  ) 
  2 
  ^ 
  Pi+P 
  2 
  =P 
  + 
  9 
  

   r\ 
  Pi 
  

  

  It 
  is 
  therefore 
  possible 
  to 
  find 
  the 
  pressures 
  from 
  the 
  curves 
  of 
  equal 
  tint 
  and 
  

   equal 
  inclination, 
  in 
  any 
  case 
  in 
  which 
  it 
  may 
  be 
  required. 
  In 
  the 
  meantime 
  

   the 
  curves 
  of 
  figs. 
  2, 
  3, 
  4 
  shew 
  the 
  correctness 
  of 
  Sir 
  John 
  Herschell's 
  ingenious 
  

   explanation 
  of 
  the 
  phenomena 
  of 
  heated 
  and 
  unannealed 
  glass. 
  

  

  Note 
  A. 
  

  

  As 
  the 
  mathematical 
  laws 
  of 
  compressions 
  and 
  pressures 
  have 
  been 
  very 
  thoroughly 
  investi- 
  

   gated, 
  and 
  as 
  they 
  are 
  demonstrated 
  with 
  great 
  elegance 
  in 
  the 
  very 
  complete 
  and 
  elaborate 
  memoir 
  

   of 
  MM. 
  Lame 
  and 
  Clapeyron, 
  I 
  shall 
  state 
  as 
  briefly 
  as 
  possible 
  their 
  results. 
  

  

  Let 
  a 
  solid 
  be 
  subjected 
  to 
  compressions 
  or 
  pressures 
  of 
  any 
  kind, 
  then, 
  if 
  through 
  any 
  point 
  in 
  

   the 
  solid 
  lines 
  be 
  drawn 
  whose 
  lengths, 
  measured 
  from 
  the 
  given 
  point, 
  are 
  proportional 
  to 
  the 
  com- 
  

   pression 
  or 
  pressure 
  at 
  the 
  point 
  resolved 
  in 
  the 
  directions 
  in 
  which 
  the 
  lines 
  are 
  drawn, 
  the 
  extre- 
  

   mities 
  of 
  such 
  lines 
  will 
  be 
  in 
  the 
  surface 
  of 
  an 
  ellipsoid, 
  whose 
  centre 
  is 
  the 
  given 
  point. 
  

  

  The 
  properties 
  of 
  the 
  system 
  of 
  compressions 
  or 
  pressures 
  may 
  be 
  deduced 
  from 
  those 
  of 
  the 
  

   ellipsoid. 
  

  

  There 
  are 
  three 
  diameters 
  having 
  perpendicular 
  ordinates, 
  which 
  are 
  called 
  the 
  principal 
  axes 
  

   of 
  the 
  ellipsoid. 
  

  

  Similarly, 
  there 
  are 
  always 
  three 
  directions 
  in 
  the 
  compressed 
  particle 
  in 
  which 
  there 
  is 
  no 
  tan- 
  

   gential 
  action, 
  or 
  tendency 
  of 
  the 
  parts 
  to 
  slide 
  on 
  one 
  another. 
  These 
  directions 
  are 
  called 
  the 
  

   principal 
  axes 
  of 
  compression 
  or 
  of 
  pressure, 
  and 
  in 
  homogeneous 
  solids 
  they 
  always 
  coincide 
  with 
  

   each 
  other. 
  

  

  The 
  compression 
  or 
  pressure 
  in 
  any 
  other 
  direction 
  is 
  equal 
  to 
  the 
  sum 
  of 
  the 
  products 
  of 
  the 
  

   compressions 
  or 
  pressures 
  in 
  the 
  principal 
  axes 
  multiplied 
  into 
  the 
  squares 
  of 
  the 
  cosines 
  of 
  the 
  

   angles 
  which 
  they 
  respectively 
  make 
  with 
  that 
  direction. 
  

  

  Note 
  B. 
  

  

  The 
  fundamental 
  equations 
  of 
  this 
  paper 
  difter 
  from 
  those 
  of 
  Navier, 
  Poisson, 
  &c, 
  only 
  in 
  not 
  

   assuming 
  an 
  invariable 
  ratio 
  between 
  the 
  linear 
  and 
  the 
  cubical 
  elasticity 
  ; 
  but 
  since 
  I 
  have 
  not 
  

   attempted 
  to 
  deduce 
  them 
  from 
  the 
  laws 
  of 
  molecular 
  action, 
  some 
  other 
  reasons 
  must 
  be 
  given 
  for 
  

   adopting 
  them. 
  

  

  The 
  experiments 
  from 
  which 
  the 
  laws 
  are 
  deduced 
  are 
  — 
  

  

  1st, 
  Elastic 
  solids 
  put 
  into 
  motion 
  vibrate 
  isochronously, 
  so 
  that 
  the 
  sound 
  does 
  not 
  vary 
  with 
  

   the 
  amplitude 
  of 
  the 
  vibrations. 
  

  

  2d, 
  Regnault's 
  experiments 
  on 
  hollow 
  spheres 
  shew 
  that 
  both 
  linear 
  and 
  cubic 
  compressions 
  

   are 
  proportional 
  to 
  the 
  pressures. 
  

  

  3d, 
  Experiments 
  on 
  the 
  elongation 
  of 
  rods 
  and 
  tubes 
  immersed 
  in 
  water, 
  prove 
  that 
  the 
  elon- 
  

   gation, 
  the 
  decrease 
  of 
  diameter, 
  and 
  the 
  increase 
  of 
  volume, 
  are 
  proportional 
  to 
  the 
  tension. 
  

  

  VOL. 
  XX. 
  PART 
  I. 
  2 
  I 
  

  

  