﻿Pressure 
  of 
  Vibrations. 
  343 
  

  

  and 
  vertical 
  coordinates 
  o£ 
  C, 
  

  

  #=Zsin0, 
  y 
  = 
  l—lcos 
  0; 
  

  

  and 
  accordingly 
  i£ 
  the 
  mass 
  o£ 
  C 
  be 
  taken 
  to 
  be 
  unity, 
  

  

  T=il 
  r2 
  (2-2cosd) 
  + 
  l 
  , 
  d'.lzm0 
  + 
  id 
  / 
  n 
  2 
  , 
  . 
  . 
  (17) 
  

  

  V, 
  & 
  denoting 
  dl/dt, 
  dd/dt. 
  Also 
  

  

  V=^(i-cos0) 
  (18) 
  

  

  From 
  (17), 
  (18) 
  

  

  d 
  ^= 
  9 
  (l-cos0), 
  ^=Wam0+d»l. 
  . 
  .(19) 
  

  

  These 
  expressions 
  are 
  general 
  ; 
  but 
  for 
  our 
  present 
  purpose 
  

   it 
  will 
  suffice 
  i£ 
  we 
  suppose 
  that 
  I' 
  is 
  zero, 
  that 
  is 
  that 
  the 
  

   ring 
  is 
  held 
  at 
  rest. 
  Accordingly 
  

  

  dV 
  = 
  Y 
  dT 
  _ 
  2T 
  

  

  dl 
  = 
  '' 
  V 
  dl 
  z 
  I 
  ' 
  

  

  and 
  (16) 
  gives 
  

  

  ( 
  Ldt 
  _ 
  1 
  j 
  

  

  'V 
  — 
  9T 
  

   —f^dt 
  (20) 
  

  

  On 
  the 
  right 
  hand 
  of 
  (20) 
  we 
  find 
  the 
  mean 
  values 
  of 
  V 
  and 
  

   and 
  of 
  T. 
  But 
  these 
  mean 
  values 
  are 
  equal. 
  In 
  fact 
  

  

  JVd*=jTrf*=p«i, 
  (21) 
  

  

  if 
  E 
  denote 
  the 
  total 
  energy. 
  Hence, 
  if 
  L 
  now 
  denote 
  the 
  

   mean 
  value, 
  

  

  L=-P/?, 
  (22) 
  

  

  the 
  negative 
  sign 
  denoting 
  that 
  the 
  mean 
  force 
  necessary 
  to 
  

   hold 
  the 
  ring 
  at 
  rest 
  must 
  be 
  applied 
  in 
  the 
  direction 
  which 
  

   tends 
  to 
  diminish 
  Z, 
  i.c 
  e. 
  downwards. 
  In 
  former 
  equations 
  

   (1), 
  (6), 
  (14), 
  L 
  had 
  the 
  reverse 
  sign. 
  

  

  We 
  will 
  now 
  consider 
  more 
  generally 
  the 
  case 
  of 
  one 
  

   dimension, 
  using 
  a 
  method 
  that 
  will 
  apply 
  equally 
  whether 
  

   for 
  example 
  the 
  vibrating 
  body 
  be 
  a 
  stretched 
  string, 
  or 
  a 
  

   rod 
  vibrating 
  flexurally. 
  All 
  that 
  we 
  postulate 
  is 
  homogeneity 
  

   of 
  constitution, 
  so 
  that 
  what 
  can 
  be 
  said 
  about 
  any 
  part 
  of 
  

   the 
  length 
  can 
  be 
  said 
  equally 
  about 
  any 
  other 
  part. 
  In 
  

   applying 
  Lagrange's 
  method 
  the 
  coordinates 
  are 
  I 
  the 
  length 
  

   of 
  the 
  vibrating 
  portion, 
  and 
  fa, 
  <£ 
  2 
  , 
  &c. 
  defining, 
  as 
  in 
  (3), 
  

   the 
  displacement 
  from 
  equilibrium 
  during 
  the 
  vibrations. 
  

   As 
  functions 
  of 
  /, 
  we 
  suppose 
  that 
  

  

  V*T, 
  TccT 
  (23) 
  

  

  