﻿118 
  Prof. 
  E. 
  H. 
  Barton 
  on 
  Range 
  and 
  

  

  under 
  sustained 
  forcing, 
  varies 
  directly 
  as 
  its 
  damping 
  coeffi- 
  

   cient 
  and 
  inversely 
  as 
  the 
  frequency 
  of 
  the 
  vibrations 
  which 
  it 
  

   is 
  sought 
  to 
  elicit 
  from 
  it. 
  

  

  And 
  of 
  the 
  various 
  ways 
  of 
  estimating 
  the 
  range, 
  this 
  

   appears 
  to 
  be 
  the 
  simplest 
  which 
  is 
  quantitative, 
  standard, 
  

   and 
  general. 
  It 
  is 
  accordingly 
  employed 
  in 
  what 
  follows, 
  

   and 
  is 
  respectfully 
  submitted 
  for 
  general 
  adoption. 
  

  

  Its 
  reciprocal, 
  the 
  diminution 
  of 
  response 
  per 
  unit 
  mistiming, 
  

   we 
  may 
  fitly 
  call 
  the 
  sharpness 
  of 
  resonance 
  (or 
  response). 
  

   Denoting 
  it 
  by 
  II, 
  we 
  have 
  

  

  H 
  =M=f 
  <*» 
  

  

  As 
  equations 
  (19) 
  and 
  (20) 
  are 
  expressed 
  very 
  compactly 
  

   by 
  symbols 
  which 
  denote 
  functions 
  of 
  other 
  quantities, 
  it 
  may 
  

   be 
  well 
  to 
  expand 
  them 
  a 
  little 
  and 
  recapitulate 
  somewhat. 
  

   "VVe 
  may 
  thus 
  write 
  

  

  ("*) 
  k 
  1 
  

  

  G= 
  \P 
  nf 
  =--=-. 
  . 
  . 
  • 
  (21) 
  

  

  In 
  this 
  equation, 
  r?/27r 
  is 
  the 
  frequency 
  of 
  the 
  impressed 
  

   force 
  (of 
  sustained 
  amplitude) 
  and 
  of 
  the 
  forced 
  vibrations, 
  

   f\2ir 
  is 
  the 
  frequency 
  that 
  the 
  natural 
  vibrations 
  of 
  the 
  

   responding 
  system 
  would 
  have 
  if 
  free 
  from 
  friction, 
  k/2 
  is 
  

   the 
  damping 
  coefficient 
  expressing 
  the 
  friction 
  actually 
  

   present 
  in 
  the 
  responding 
  system, 
  N 
  is 
  the 
  kinetic 
  energy 
  

   of 
  the 
  responding 
  system 
  (at 
  the 
  instant 
  when 
  its 
  potential 
  

   energy 
  vanishes) 
  per 
  unit 
  value 
  of 
  the 
  mean 
  square 
  of 
  the 
  

   impressed 
  force, 
  N 
  is 
  the 
  maximum 
  value 
  attained 
  by 
  N 
  

   and 
  occurs 
  for 
  n=:p. 
  It 
  must 
  be 
  carefully 
  borne 
  in 
  mind 
  

   that 
  these 
  simple 
  values 
  of 
  range 
  and 
  sharpness 
  are 
  only 
  

   valid 
  for 
  the 
  resonance 
  under 
  sustained 
  forcing. 
  

  

  We 
  may 
  now 
  inquire 
  what 
  are 
  the 
  dimensions, 
  if 
  any, 
  of 
  

   the 
  range 
  and 
  sharpness 
  of 
  resonance. 
  It 
  is 
  easily 
  seen 
  that 
  

   D 
  and 
  M 
  are 
  each 
  of 
  no 
  dimensions, 
  so 
  that 
  their 
  quotients 
  

   G 
  and 
  H 
  must 
  be 
  pure 
  numbers 
  also. 
  If 
  the 
  equations 
  (1), 
  

   (2), 
  (3), 
  and 
  (4) 
  are 
  referred 
  to, 
  we 
  find 
  that, 
  both 
  in 
  their 
  

   mechanical 
  and 
  electrical 
  uses, 
  k 
  and 
  p 
  are 
  each 
  of 
  dimensions 
  

   minus-one 
  in 
  time. 
  This 
  shows 
  again 
  that 
  their 
  quotients 
  are 
  

   pure 
  numbers. 
  

  

  If 
  preferred, 
  the 
  range 
  and 
  sharpness 
  of 
  resonance 
  may 
  be 
  

   regarded 
  as 
  the 
  tangent 
  and 
  cotangent 
  respectively 
  of 
  an 
  

   angle, 
  <f> 
  say, 
  which 
  may 
  be 
  called 
  the 
  angle 
  of 
  resonance. 
  

  

  