﻿concerning 
  Forced 
  Vibrations 
  and 
  Resonance. 
  115 
  

  

  We 
  now 
  suppose 
  that 
  Mod- 
  M* 
  is 
  constant, 
  while 
  p 
  varies 
  

   over 
  the 
  small 
  range 
  in 
  the 
  neighbourhood 
  of 
  n 
  for 
  which 
  

   alone 
  Mod 
  2 
  yjr 
  is 
  sensible, 
  and 
  inquire 
  as 
  to 
  the 
  sum 
  of 
  the 
  

   values 
  of 
  Mod 
  yfr. 
  Since 
  

  

  r^-' 
  <*» 
  

  

  —♦*-=¥*; 
  

  

  or 
  again 
  

  

  we 
  find 
  

  

  I 
  

  

  ic 
  ("Mod 
  2 
  yfr 
  dp 
  = 
  ~ 
  h 
  Mod 
  2 
  ¥ 
  (84) 
  

  

  On 
  the 
  left 
  \c 
  Mod 
  2 
  yfr 
  represents 
  the 
  potential 
  energy 
  of 
  the 
  

   system 
  at 
  the 
  phase 
  of 
  maximum 
  displacement, 
  which 
  is 
  

   the 
  same 
  as 
  the 
  nearly 
  constant 
  total 
  energy, 
  so 
  that 
  (84) 
  

   gives 
  the 
  integral 
  of 
  this 
  total 
  energy 
  as 
  p 
  passes 
  through 
  the 
  

   value 
  which 
  calls 
  out 
  the 
  maximum 
  and 
  (by 
  supposition) 
  very 
  

   great 
  resonance. 
  

  

  The 
  most 
  remarkable 
  feature 
  of 
  (84) 
  is 
  perhaps 
  that 
  the 
  

   integral 
  is 
  independent 
  of 
  a 
  and 
  c. 
  Large 
  values 
  of 
  these 
  

   quantities 
  will 
  increase 
  the 
  energy 
  of 
  the 
  system 
  at 
  the 
  point 
  

   where 
  p 
  = 
  n; 
  but 
  on 
  the 
  other 
  hand 
  this 
  maximum 
  falls 
  off 
  

   more 
  rapidly 
  as 
  p 
  departs 
  from 
  the 
  special 
  value. 
  

  

  We 
  pass 
  next 
  to 
  the 
  more 
  difficult 
  considerations 
  which 
  

   arise 
  when 
  the 
  force 
  ^t 
  2 
  is 
  of 
  one 
  kind, 
  while 
  the 
  coordinate 
  

   •tyi 
  on 
  which 
  the 
  resonance 
  principally 
  depends 
  is 
  of 
  another. 
  

   In 
  the 
  first 
  instance 
  we 
  shall 
  suppose 
  that 
  there 
  are 
  no 
  other 
  

   than 
  these 
  two 
  degrees 
  of 
  freedom. 
  

  

  If 
  in 
  equation 
  (3) 
  we 
  assume 
  SE^, 
  ^r 
  3 
  , 
  yjr±, 
  &c. 
  to 
  vanish, 
  

   we 
  get 
  

  

  *2 
  ^12 
  — 
  e 
  \l 
  e 
  2'2 
  

  

  where 
  e 
  n 
  , 
  e 
  12 
  , 
  e 
  22 
  have 
  the 
  values 
  given 
  in 
  (4) 
  with 
  ip 
  sub- 
  

   stituted 
  for 
  D. 
  We 
  suppose 
  further 
  that 
  6 
  12 
  =0, 
  b 
  n 
  = 
  0, 
  so 
  

   that 
  the 
  dissipation 
  depends 
  entirely 
  on 
  b 
  22 
  . 
  With 
  these 
  

   simplifications 
  the 
  numerator 
  of 
  (85) 
  becomes 
  

  

  ^i2= 
  c 
  i2— 
  i? 
  2 
  «i2, 
  (86) 
  

  

  and 
  for 
  the 
  denominator 
  (taken 
  negatively) 
  

  

  *11*22 
  - 
  ?U 
  = 
  fall 
  — 
  /tfll) 
  (t'22 
  — 
  P 
  2 
  0>22) 
  

  

  ~(^2-^ 
  2 
  %2) 
  2 
  + 
  ^^(cn-/> 
  2 
  «ii).. 
  • 
  • 
  (87) 
  

   12 
  

  

  