﻿112 
  Prof. 
  E. 
  H. 
  Barton 
  on 
  Range 
  and 
  

  

  that 
  the 
  diminution 
  of 
  resonance 
  for 
  a 
  given 
  mistiming 
  

   depends 
  upon 
  the 
  damping 
  of 
  the 
  responding 
  system. 
  The 
  

   resonance 
  is 
  said 
  to 
  be 
  sharper 
  when 
  this 
  damping 
  is 
  small. 
  

   Or, 
  we 
  might 
  say 
  that 
  the 
  range 
  of 
  resonance 
  is 
  greater 
  when 
  

   the 
  damping 
  is 
  large. 
  

  

  This 
  dependence 
  of 
  sharpness 
  of 
  resonance 
  on 
  damping 
  is 
  

   explained 
  in 
  the 
  mathematical 
  treatment 
  of 
  forced 
  vibrations, 
  

   and 
  is 
  often 
  touched 
  upon 
  even 
  in 
  semi-popular 
  descriptions. 
  

  

  There 
  appears, 
  however, 
  to 
  be 
  another 
  factor 
  affecting 
  the 
  

   range 
  of 
  resonance, 
  viz. 
  the 
  pitch 
  or 
  frequency 
  of 
  the 
  vibrations 
  

   which 
  one 
  endeavours 
  to 
  elicit. 
  And, 
  when 
  the 
  harmonic 
  

   impressed 
  forces 
  are 
  sustained, 
  this 
  second 
  factor 
  is 
  just 
  as 
  

   fundamental 
  and 
  inevitable 
  in 
  its 
  influence 
  as 
  the 
  first. 
  But, 
  

   whereas 
  the 
  effect 
  of 
  the 
  damping 
  is 
  familiar 
  to 
  all 
  students 
  

   of 
  forced 
  vibrations, 
  that 
  of 
  the 
  pitch 
  seems, 
  up 
  to 
  now, 
  to 
  

   have 
  escaped 
  all 
  widespread 
  recognition, 
  if 
  even 
  it 
  has 
  been 
  

   understood 
  by 
  a 
  few. 
  Yet 
  its 
  theoretical 
  derivation 
  and 
  

   expression 
  are 
  of 
  the 
  simplest. 
  For 
  it 
  is 
  implied 
  in 
  any 
  of 
  

   the 
  equations 
  used 
  as 
  a 
  solution 
  for 
  vibrations 
  under 
  sustained 
  

   forcing. 
  Accordingly 
  these 
  only 
  need 
  rightly 
  handling 
  for 
  

   the 
  clear 
  exhibition 
  of 
  the 
  relation 
  which 
  apparently 
  has 
  been 
  

   so 
  often 
  overlooked. 
  

  

  It 
  will 
  thus 
  be 
  shown 
  that 
  the 
  range 
  of 
  resonance, 
  which 
  

   varies 
  directly 
  as 
  the 
  damping 
  coefficient, 
  also 
  varies 
  inversely 
  

   as 
  the 
  frequency. 
  

  

  The 
  application 
  of 
  this 
  principle 
  extends 
  to 
  any 
  cases 
  of 
  

   sustained 
  forced 
  vibrations 
  whether 
  mechanical, 
  acoustical, 
  

   or 
  electrical. 
  

  

  The 
  object 
  of 
  the 
  present 
  paper 
  is 
  to 
  sketch 
  the 
  necessary 
  

   theory 
  and 
  illustrate 
  it 
  by 
  a 
  few 
  experiments. 
  These, 
  though 
  

   apparently 
  simple, 
  are 
  often 
  somewhat 
  complicated, 
  and 
  need 
  

   this 
  principle 
  as 
  a 
  clue 
  to 
  their 
  explanation 
  or 
  more 
  suitable 
  

   arrangement. 
  It 
  will 
  thus 
  be 
  found 
  that 
  the 
  principle 
  under 
  

   discussion 
  throws 
  light 
  upon 
  some 
  apparently 
  anomalous 
  notes 
  

   obtainable 
  on 
  certain 
  brass 
  instruments. 
  

  

  It 
  may 
  be 
  noted 
  here 
  that 
  several 
  writers 
  have 
  shown 
  that 
  

   the 
  character 
  of 
  the 
  resonance 
  is 
  a 
  function 
  of 
  the 
  logarithmic 
  

   decrement 
  of 
  the 
  responding 
  system. 
  But 
  the 
  decrement 
  

   appears 
  to 
  have 
  been 
  treated 
  as 
  a 
  single 
  quantity 
  measuring 
  

   the 
  damping. 
  The 
  possibility 
  of 
  changing 
  the 
  logarithmic 
  

   decrement, 
  and 
  therefore 
  the 
  sharpness 
  of 
  resonance, 
  by 
  a 
  

   change 
  in 
  frequency, 
  the 
  damping 
  coefficient 
  remaining 
  

   constant, 
  seems 
  to 
  have 
  escaped 
  their 
  attention. 
  

  

  