﻿458 
  Dr. 
  J. 
  Morrow 
  on 
  the 
  Lateral 
  Deflexion 
  and 
  

  

  Now, 
  when 
  PP/EI 
  is 
  small, 
  6 
  differs 
  little 
  from 
  2-36502, 
  

   and 
  we 
  have, 
  to 
  the 
  first 
  order 
  in 
  this 
  difference 
  

  

  <^ 
  = 
  M 
  +rgl 
  (^-2-365), 
  

  

  that 
  is 
  

  

  aZ 
  = 
  10-5083 
  -3-M32/3/ 
  (13) 
  

  

  Eliminating 
  « 
  from 
  (12) 
  and 
  (13), 
  we 
  find, 
  after 
  an 
  expan- 
  

   sion 
  by 
  the 
  Binomial 
  as 
  far 
  as 
  the 
  second 
  term 
  

  

  ^2/2 
  ^5.5933__ 
  .9251^, 
  

  

  by 
  means 
  of 
  which 
  the 
  second 
  of 
  (10) 
  gives 
  

  

  ^^ 
  = 
  21-2«%^*{l 
  + 
  -0983l|f-00557(gJ}. 
  (14) 
  

  

  § 
  12. 
  For 
  harmonics 
  we 
  may 
  in 
  all 
  cases 
  take 
  

   tanh 
  (^ 
  = 
  1 
  ; 
  

   thus, 
  reasoning 
  as 
  in 
  the 
  last 
  paragraph, 
  

  

  V= 
  (2i-i)7r, 
  

   1 
  PP 
  

  

  where 
  

   whence 
  

  

  /3'P 
  = 
  iv'- 
  

  

  EI 
  

  

  and 
  

  

  ^' 
  - 
  (2 
  ) 
  ^^ 
  "^ 
  1 
  (1 
  ""-^ 
  JpoyP 
  "" 
  V~P^EI 
  • 
  ^^^^ 
  

  

  Section 
  lY. 
  — 
  Massive 
  Bar, 
  Load 
  at 
  Directed 
  End. 
  

  

  § 
  13. 
  When 
  a 
  massive 
  clamped-directed 
  bar 
  carries 
  a 
  load 
  

   concentrated 
  at 
  its 
  directed 
  end, 
  the 
  frequency 
  and 
  type 
  of 
  

   lateral 
  vibration 
  can 
  be 
  found 
  to 
  any 
  required 
  degree 
  of 
  

   accuracy 
  by 
  the 
  method 
  of 
  continuous 
  approximation 
  *. 
  

  

  Fig. 
  4. 
  

  

  DC 
  ^ 
  

  

  ^-^Uijf/^ 
  

  

  In 
  figure 
  4, 
  let 
  the 
  origin 
  be 
  at 
  the 
  clamped 
  end 
  and 
  

   * 
  See 
  Phil. 
  Mag., 
  July 
  1905, 
  pp. 
  113-125, 
  and 
  March 
  1906, 
  pp. 
  351-374. 
  

  

  