﻿Sharpness 
  of 
  Resonance 
  under 
  Sustained 
  Forcing. 
  113 
  

   Mathematical 
  Theory. 
  

  

  Equations 
  for 
  Forced 
  Vibrations. 
  — 
  Let 
  a 
  particle 
  of 
  mass 
  m 
  

   of 
  the 
  responding 
  system 
  have 
  a 
  restoring 
  force 
  s 
  per 
  unit 
  

   displacement 
  along 
  the 
  axis 
  of 
  i/, 
  resistance 
  r 
  per 
  unit 
  velocity, 
  

   and 
  be 
  under 
  the 
  action 
  of 
  a 
  damped 
  harmonic 
  impressed 
  

   force 
  Fe~ 
  ht 
  sin 
  nt. 
  

  

  Then 
  the 
  equation 
  of 
  motion 
  may 
  be 
  written 
  

  

  >»yf 
  +r£ 
  + 
  si/ 
  = 
  ¥e~ 
  ht 
  sin 
  nt. 
  . 
  . 
  . 
  (1) 
  

  

  Though 
  we 
  shall 
  be 
  concerned 
  chiefly 
  with 
  sustained 
  

   harmonic 
  forces, 
  it 
  is 
  well 
  to 
  begin 
  with 
  damped 
  ones 
  and 
  

   note 
  the 
  distinction, 
  as 
  this 
  is 
  required 
  for 
  the 
  elucidation 
  of 
  

   some 
  of 
  the 
  experiments. 
  

  

  Further, 
  as 
  reference 
  will 
  be 
  made 
  to 
  electrical 
  vibrations, 
  

   it 
  is 
  desirable 
  to 
  make 
  the 
  work 
  more 
  general 
  by 
  writing 
  here 
  

   the 
  equation 
  of 
  motion 
  applicable 
  to 
  them 
  also. 
  Thus, 
  if 
  a 
  

   quantity 
  Q 
  of 
  electricity 
  is 
  on 
  one 
  plate 
  of 
  a 
  condenser 
  of 
  

   capacity 
  C 
  which 
  is 
  connected 
  to 
  the 
  other 
  plate 
  by 
  a 
  circuit 
  

   of 
  resistance 
  R 
  and 
  inductance 
  L, 
  the 
  system 
  being 
  under 
  

   the 
  damped 
  harmonic 
  forcing 
  Fle~ 
  u 
  sin 
  nt, 
  we 
  may 
  write 
  

  

  L 
  S 
  +R 
  f 
  + 
  §=e«- 
  m 
  -«*, 
  • 
  • 
  • 
  (2) 
  

  

  provided 
  that 
  the 
  mutual 
  inductance 
  between 
  the 
  circuits 
  is 
  

   small. 
  

  

  Dividing 
  each 
  of 
  these 
  equations 
  by 
  the 
  coefficient 
  of 
  its 
  

   first 
  term 
  and 
  then 
  introducing 
  new 
  symbols 
  for 
  the 
  quotients 
  

   thus 
  arising, 
  we 
  reduce 
  them 
  to 
  the 
  common 
  form 
  : 
  — 
  

  

  '63 
  

  

  § 
  + 
  *f 
  +fy=fe- 
  u 
  sin 
  nt,. 
  ... 
  (3) 
  

  

  where 
  jj 
  = 
  a 
  displacement 
  of 
  a 
  particle, 
  

   or 
  a 
  quantity 
  of 
  electricity, 
  

  

  7 
  r 
  R 
  

  

  £=-, 
  or 
  =- 
  

  

  m 
  

  

  s 
  

   p-= 
  , 
  

   m 
  

  

  or 
  

  

  1 
  

  

  w 
  

  

  , 
  F 
  E 
  

  

  m 
  

  

  j; 
  

  

  0) 
  

  

  The 
  solution 
  of 
  (3) 
  is 
  well 
  known 
  to 
  be 
  the 
  sum 
  of 
  the 
  

   Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  26. 
  No. 
  151. 
  July 
  1913. 
  I 
  

  

  