﻿122 
  Prof. 
  E. 
  H. 
  Barton 
  on 
  Range 
  and 
  

  

  but 
  can 
  observe 
  and 
  measure 
  G 
  for 
  various 
  values 
  of 
  p 
  ; 
  then 
  

   the 
  variation 
  of 
  k 
  may 
  be 
  inferred 
  from 
  the 
  theoretical 
  

   relation 
  between 
  it 
  and 
  G. 
  In 
  this 
  case, 
  an 
  indirect 
  

   determination 
  would 
  be 
  made 
  for 
  k 
  where 
  a 
  direct 
  one 
  was 
  

   not 
  available. 
  

  

  Ilelmholtz's 
  Theory 
  for 
  the 
  Ear. 
  — 
  In 
  one 
  of 
  the 
  appendices 
  

   of 
  his 
  classic 
  work, 
  Helmholtz 
  mathematically 
  treated 
  the 
  

   case 
  of 
  forced 
  vibrations, 
  in 
  order 
  to 
  throw 
  light 
  upon 
  

   the 
  damping 
  of 
  the 
  responsive 
  portions 
  of 
  the 
  human 
  ear. 
  

   (See 
  'Die 
  Lehre 
  von 
  der 
  Tonempfindungen, 
  &c/, 
  BeilagelX. 
  

   and 
  pp. 
  213-217, 
  Braunschweig, 
  1865 
  ; 
  or 
  Ellis's 
  English 
  

   translation, 
  ' 
  Sensations 
  of 
  Tone, 
  &c.', 
  Appendix 
  X. 
  and 
  

   pp. 
  142-145, 
  London, 
  1895.) 
  

  

  In 
  the 
  corresponding 
  part 
  of 
  the 
  text 
  he 
  gave 
  in 
  a 
  table 
  

   some 
  numerical 
  results 
  of 
  the 
  theory 
  over 
  the 
  range 
  that 
  he 
  

   considered 
  relevant 
  to 
  the 
  case 
  under 
  discussion. 
  But 
  he 
  

   formulated 
  no 
  quantitative 
  definition 
  of 
  sharpness 
  of 
  reso- 
  

   nance 
  or 
  of 
  its 
  range, 
  neither 
  did 
  he 
  point 
  out 
  that 
  these 
  

   quantities 
  might 
  vary 
  with 
  pitch. 
  On 
  the 
  contrary 
  his 
  

   entire 
  emphasis 
  was 
  placed 
  on 
  the 
  damping. 
  His 
  method, 
  

   however, 
  when 
  examined, 
  is 
  found 
  to 
  involve 
  the 
  dependence 
  

   of 
  sharpness 
  of 
  resonance 
  on 
  what 
  he 
  regarded 
  as 
  approxi- 
  

   mately 
  the 
  logarithmic 
  decrement. 
  And 
  this 
  quantity 
  is 
  really 
  

   the 
  quotient 
  k\p 
  found 
  in 
  this 
  paper 
  as 
  the 
  value 
  of 
  the 
  true 
  

   range 
  G, 
  see 
  equations 
  (25) 
  and 
  (26). 
  Indeed, 
  at 
  a 
  certain 
  

   intermediate 
  stage 
  in 
  an 
  explanation 
  of 
  HelmhohVs 
  treat- 
  

   ment, 
  given 
  by 
  the 
  present 
  writer 
  (' 
  Text-Book 
  of 
  Sound/ 
  

   Equation 
  (18), 
  p. 
  148, 
  London, 
  1908), 
  an 
  expression 
  was 
  

   used 
  which 
  is 
  equivalent 
  to 
  equation 
  (21) 
  of 
  this 
  paper, 
  the 
  

   one 
  being 
  the 
  inversion 
  of 
  the 
  other. 
  But 
  the 
  earlier 
  

   equation 
  was 
  written 
  purely 
  as 
  a 
  paraphrase 
  of 
  Helmholtz 
  

   without 
  the 
  faintest 
  idea 
  that 
  it 
  expressed 
  anything 
  more 
  

   than 
  the 
  part 
  played 
  by 
  the 
  damping 
  which 
  was 
  then 
  re- 
  

   ceiving 
  attention. 
  Its 
  full 
  significance 
  was 
  not 
  realized 
  

   until 
  it 
  had 
  been 
  independently 
  obtained 
  in 
  the 
  manner 
  

   here 
  set 
  forth, 
  and 
  the 
  likeness 
  between 
  the 
  two 
  accidentally 
  

   noticed. 
  

  

  In 
  the 
  numerical 
  work 
  referred 
  to, 
  Helmholtz 
  took 
  for 
  

   each 
  case 
  that 
  particular 
  mistuning 
  which 
  involved 
  a 
  re- 
  

   duction 
  of 
  the 
  kinetic 
  energy 
  of 
  response 
  to 
  one-tenth 
  of 
  its 
  

   maximum. 
  He 
  also 
  gauged 
  the 
  logarithmic 
  decrement 
  of 
  

   the 
  responding 
  system 
  by 
  the 
  number 
  (x 
  say) 
  of 
  complete 
  

   periods 
  required 
  for 
  the 
  energy 
  of 
  the 
  natural 
  vibrations 
  to 
  

   sink 
  to 
  one-tenth 
  of 
  its 
  initial 
  value. 
  But 
  the 
  estimate 
  of 
  

   this 
  number 
  {x) 
  was 
  made 
  on 
  the 
  approximate 
  supposition 
  

   that 
  the 
  period 
  was 
  unmodified 
  by 
  friction. 
  

  

  