﻿126 
  Prof. 
  E. 
  H. 
  Barton 
  on 
  Range 
  and 
  

  

  Hence, 
  when 
  this 
  damping 
  is 
  very 
  small, 
  the 
  beats 
  are 
  

   correspondingly 
  persistent. 
  

  

  The 
  disturbance 
  o£ 
  persistent 
  beats 
  is 
  reduced 
  by 
  making 
  

   the 
  lower 
  bob 
  of 
  less 
  dense 
  material 
  than 
  the 
  upper. 
  Thus 
  

   the 
  upper 
  bob 
  may 
  be 
  of 
  solid 
  iron 
  in 
  order 
  that 
  its 
  damping 
  

   shall 
  be 
  negligible, 
  and 
  the 
  lower 
  bob 
  of 
  hollow 
  celluloid 
  to 
  

   make 
  its 
  damping 
  large 
  and 
  thus 
  favour 
  tuning 
  and 
  avoid 
  

   such 
  persistent 
  beating. 
  

  

  There 
  is 
  yet 
  another 
  undesirable 
  complication 
  in 
  the 
  double 
  

   pendulum 
  which 
  is 
  not 
  provided 
  for 
  in 
  the 
  simple 
  equations. 
  

   This 
  arises 
  from 
  the 
  fact 
  that 
  the 
  period 
  of 
  a 
  simple 
  pendulum 
  

   is 
  dependent 
  slightly 
  on 
  the 
  amplitude, 
  the 
  value 
  commonly 
  

   assigned 
  being 
  a 
  lower 
  limit 
  which 
  is 
  approached 
  as 
  the 
  

   amplitude 
  vanishes. 
  Thus, 
  if 
  the 
  force 
  is 
  almost 
  exactly 
  in 
  

   tune 
  with 
  the 
  small 
  oscillations 
  of 
  a 
  pendulum, 
  this 
  leads 
  to 
  

   larger 
  amplitudes 
  which 
  require 
  a 
  somewhat 
  slower 
  forcing, 
  

   and 
  so 
  their 
  further 
  increase 
  is 
  checked. 
  

  

  Accordingly, 
  if 
  the 
  simple 
  equations 
  are 
  to 
  apply 
  to 
  and 
  

   receive 
  illustration 
  from 
  the 
  double 
  pendulum, 
  its 
  vibrations 
  

   should 
  be 
  confined 
  within 
  those 
  limits 
  of 
  amplitude 
  which 
  do 
  

   not 
  appreciably 
  affect 
  the 
  period. 
  

  

  Observing 
  all 
  the 
  above 
  precautions 
  with 
  the 
  individual 
  

   pendulums, 
  each 
  about 
  a 
  metre 
  long, 
  the 
  beats 
  were 
  still 
  

   persistent, 
  so 
  that 
  no 
  steady 
  forced 
  vibration 
  could 
  be 
  

   maintained. 
  Thus, 
  in 
  changing 
  the 
  length 
  of 
  the 
  lower 
  

   pendulum 
  by 
  about 
  a 
  centimetre 
  each 
  time 
  and 
  observing 
  

   more 
  than 
  a 
  dozen 
  such 
  settings, 
  the 
  phenomena 
  of 
  beats 
  

   always 
  appeared 
  and 
  brought 
  the 
  amplitude 
  to 
  zero 
  after 
  

   from 
  10 
  to 
  12 
  periods 
  of 
  the 
  pendulum. 
  

  

  The 
  lower 
  pendulum 
  was 
  then 
  hung 
  from 
  a 
  fixed 
  point, 
  

   set 
  to 
  various 
  lengths, 
  the 
  successive 
  amplitudes 
  in 
  each 
  case 
  

   observed, 
  the 
  logarithmic 
  decrement 
  and 
  the 
  damping 
  

   coefficient 
  calculated. 
  The 
  latter 
  proved 
  to 
  be 
  practically 
  

   independent 
  of 
  period 
  and 
  of 
  value 
  expressed 
  by 
  & 
  = 
  0*075, 
  

   for 
  the 
  bob 
  of 
  a 
  thin 
  spherical 
  shell 
  of 
  celluloid 
  6 
  cm. 
  diam. 
  

   (It 
  was 
  a 
  toy 
  ball 
  of 
  the 
  sort 
  formerly 
  used 
  for 
  parlour 
  

   tennis 
  or 
  " 
  ping 
  pong.") 
  Thus, 
  since 
  k 
  was 
  found 
  constant, 
  

   the 
  range 
  of 
  resonance 
  Gr 
  might 
  be 
  expected 
  to 
  vary 
  

   inversely 
  as 
  the 
  frequency, 
  i. 
  e., 
  directly 
  as 
  the 
  square 
  root 
  

   of 
  the 
  length 
  of 
  the 
  pendulum. 
  Hence, 
  with 
  longer 
  

   pendulums, 
  the 
  resonance 
  should 
  be 
  less 
  sharp 
  and 
  it 
  should 
  

   be 
  easier 
  to 
  attain 
  a 
  tuning 
  sufficiently 
  exact 
  to 
  banish 
  the 
  

   beats. 
  

  

  For 
  example, 
  with 
  a 
  four-fold 
  length, 
  which 
  was 
  then 
  

   adopted, 
  the 
  period 
  and 
  range 
  of 
  resonance 
  were 
  each 
  

  

  