﻿454 
  Dr. 
  J. 
  Morrow 
  on 
  the 
  Lateral 
  Deflexion 
  and 
  

  

  end, 
  cc 
  = 
  l, 
  di/ldx 
  = 
  0, 
  and, 
  i£ 
  there 
  be 
  no 
  concentrated 
  load 
  

   there, 
  d^yldx^=0. 
  

  

  Section 
  TI. 
  — 
  Unloaded 
  Massive 
  Bar. 
  

   § 
  4. 
  In 
  the 
  ordinary 
  case 
  of 
  a 
  bar 
  vibrating 
  under 
  the 
  

   effect 
  of 
  its 
  own 
  mass 
  only, 
  the 
  form 
  of 
  the 
  elastic 
  central 
  

   line 
  and 
  the 
  frequency 
  of 
  the 
  lateral 
  vibrations 
  are 
  to 
  be 
  

   determined 
  from 
  the 
  well-known 
  equation 
  

  

  in 
  which 
  

  

  d'y 
  

  

  

  and 
  of 
  which 
  the 
  general 
  solution 
  is 
  

  

  y 
  = 
  A 
  sin 
  fix 
  + 
  B 
  cos 
  fjux 
  + 
  C 
  sinli 
  fjux 
  4- 
  D 
  cosh 
  ^x. 
  

  

  The 
  end 
  conditions 
  give 
  four 
  equations 
  whose 
  compatibility 
  

   requires 
  that 
  

  

  tan 
  /jlI= 
  — 
  tanh 
  /xlj 
  .... 
  (3) 
  

  

  of 
  which 
  the 
  least 
  root 
  is 
  

  

  fMl 
  = 
  2-36b02 
  (4) 
  

  

  The 
  frequency 
  of 
  the 
  fundamental 
  is 
  therefore 
  

  

  5-5933 
  /ET 
  

  

  § 
  5. 
  It 
  can 
  be 
  shown 
  that 
  the 
  values 
  of 
  satisfying 
  

  

  cos 
  ^ 
  cosh 
  ^=1, 
  (5) 
  

  

  are 
  the 
  same 
  as 
  those 
  which 
  satisfy 
  

  

  tan 
  -r 
  = 
  -h 
  tanh 
  — 
  

   2 
  " 
  2 
  ' 
  

  

  and 
  hence 
  that 
  the 
  admissible 
  roots 
  of 
  equation 
  (3) 
  are 
  each 
  

   one 
  half 
  of 
  the 
  alternate 
  roots 
  of 
  (5). 
  Accurate 
  solutions 
  

   of 
  (5) 
  have 
  been 
  provided 
  by 
  Rayleigh 
  ^, 
  and 
  from 
  these 
  the 
  

   values 
  of 
  fxl 
  which 
  satisfy 
  (3) 
  are 
  

  

  yu,iZ 
  = 
  2-365020 
  

  

  yL^2^ 
  = 
  5-497804 
  

  

  /^3/ 
  = 
  8-639380, 
  

   after 
  which 
  

  

  /jlJ= 
  (n~i)7r, 
  

  

  to 
  more 
  than 
  six 
  decimal 
  places, 
  n 
  being 
  an 
  integer. 
  

  

  * 
  ' 
  Theory 
  of 
  Sound,' 
  Eayleigli; 
  vol. 
  i. 
  p. 
  278, 
  1894 
  edition. 
  

  

  