﻿Sharpness 
  of 
  Resonance 
  under 
  Sustained 
  Forcing. 
  117 
  

  

  the 
  sum 
  of: 
  measures 
  of 
  separate 
  intervals 
  equal 
  to 
  the 
  measure 
  

   of 
  the 
  whole 
  or 
  resultant 
  interval. 
  

  

  It 
  may 
  be 
  noted 
  that 
  if 
  the 
  interval 
  n/p 
  be 
  called 
  I 
  loga- 
  

   rithmic 
  cents, 
  of 
  which 
  1200 
  make 
  the 
  octave, 
  we 
  have 
  

  

  log 
  2 
  c 
  V/>/ 
  

   whence 
  1 
  = 
  867 
  31 
  nearly, 
  for 
  small 
  intervals. 
  

  

  (IGa) 
  

  

  The 
  values 
  of 
  n/p 
  and 
  I 
  are 
  indicated 
  along 
  the 
  base 
  of 
  

   fig. 
  1, 
  below 
  the 
  values 
  of 
  the 
  mistiming 
  M. 
  

  

  Resonance, 
  its 
  Range 
  and 
  Sharpness. 
  — 
  By 
  reference 
  to 
  any 
  

   of 
  the 
  resonance 
  curves 
  in 
  fig. 
  1, 
  it 
  is 
  evident 
  that 
  the 
  mis- 
  

   tuning 
  allowable 
  for 
  any 
  given 
  value 
  of 
  resonance 
  measures 
  

   in 
  a 
  certain 
  way 
  the 
  range 
  of 
  resonance 
  under 
  the 
  circum- 
  

   stances 
  represented 
  by 
  that 
  curve. 
  But 
  since 
  this 
  degree 
  of 
  

   resonance 
  may 
  be 
  chosen 
  more 
  or 
  less 
  arbitrarily, 
  it 
  is 
  difficult 
  

   to 
  found 
  on 
  this 
  plan 
  a 
  standard 
  measure 
  of 
  the 
  range. 
  

  

  Further, 
  the 
  measurement 
  must 
  be 
  such 
  that 
  the 
  range 
  

   derived 
  from 
  a 
  given 
  resonance 
  curve 
  would 
  have 
  the 
  same 
  

   value 
  whether 
  that 
  range 
  of 
  resonance 
  were 
  estimated: 
  — 
  

  

  (i.) 
  As 
  directly 
  proportional 
  to 
  the 
  mistiming 
  allowable 
  for 
  

   any 
  given 
  diminution 
  of 
  response, 
  or 
  

  

  (ii.) 
  As 
  inversely 
  proportional 
  to 
  the 
  diminution 
  of 
  response 
  

   involved 
  by 
  any 
  given 
  mistiming. 
  

  

  And 
  the 
  method 
  of 
  estimating 
  the 
  diminution 
  of 
  response 
  

   to 
  make 
  the 
  second 
  alternative 
  harmonize 
  with 
  the 
  first 
  

   is 
  in 
  no 
  wise 
  arbitrary, 
  being 
  precisely 
  dictated 
  by 
  the 
  curve 
  

   itself, 
  whose 
  coordinates 
  are 
  the 
  M 
  and 
  N 
  of 
  equations 
  (12) 
  

   and 
  {I'd). 
  This 
  method 
  may 
  accordingly 
  serve 
  to 
  measure 
  

   the 
  range 
  in 
  a 
  standard 
  manner. 
  Following 
  this 
  clue, 
  we 
  

   find 
  from 
  equations 
  (12) 
  and 
  (15) 
  

  

  - 
  f 
  - 
  = 
  S 
  - 
  wk*)pv 
  + 
  *>-iF 
  , 
  - 
  D 
  '^' 
  (l7) 
  

  

  where 
  D=V(N 
  -N)-t-N, 
  (18) 
  

  

  and 
  may 
  be 
  called 
  the 
  diminution 
  of 
  resonance 
  or 
  response 
  

   corresponding 
  to 
  the 
  mistuning 
  M. 
  

  

  Hence, 
  if 
  G 
  denotes 
  the 
  range 
  of 
  resonance, 
  defined 
  as 
  the 
  

   (Quotient 
  of 
  mistuning 
  and 
  the 
  corresponding 
  diminution 
  of 
  

   response, 
  we 
  have 
  from 
  equation 
  (17) 
  

  

  G=}£=*. 
  ...... 
  . 
  (19) 
  

  

  Or, 
  in 
  other 
  words, 
  the 
  range 
  of 
  resonance 
  of 
  a 
  system 
  

  

  