﻿Coupled 
  Circuits 
  and 
  Mechanical 
  Analogies. 
  205 
  

  

  the 
  coefficients 
  and 
  their 
  physical 
  significance. 
  Thus 
  in 
  (3) 
  

   for 
  the 
  electrical 
  case 
  it 
  will 
  be 
  seen 
  that 
  the 
  same 
  coefficient 
  

   occurs 
  on 
  the 
  right 
  side 
  of 
  each 
  equation. 
  This 
  is 
  M 
  the 
  

   coefficient 
  of 
  mutual 
  induction. 
  Also 
  the 
  first 
  coefficients 
  at 
  

   the 
  left 
  of 
  each 
  equation 
  respectively 
  are 
  L 
  and 
  N 
  the 
  separate 
  

   self-inductions. 
  Now 
  this 
  correspondence 
  of 
  coefficients 
  does 
  

   not 
  hold 
  between 
  (3) 
  and 
  (2) 
  but 
  does 
  hold 
  between 
  (3) 
  and 
  

   (1). 
  And 
  for 
  this 
  reason 
  the 
  form 
  (1) 
  is 
  probably 
  preferable 
  

   to 
  the 
  experimental 
  physicist, 
  though 
  the 
  other 
  form 
  (2) 
  

   is 
  distinctly 
  illuminating 
  and 
  may 
  be 
  preferred 
  by 
  the 
  

   mathematician 
  . 
  

  

  4. 
  That 
  the 
  pendulums 
  represented 
  by 
  equations 
  (1) 
  and 
  

   (2) 
  are 
  not 
  in 
  the 
  complete 
  sense 
  an 
  exact 
  analogy 
  to 
  the 
  

   electrical 
  case 
  of 
  (3) 
  may 
  also 
  be 
  seen 
  from 
  the 
  relations 
  of 
  

   the 
  frequencies 
  of 
  the 
  coupled 
  vibrations 
  in 
  the 
  two 
  cases. 
  

   Suppose 
  the 
  two 
  separate 
  vibrations 
  for 
  pendulums 
  or 
  those 
  

   for 
  the 
  electrical 
  circuits 
  to 
  be 
  equal 
  and 
  denote 
  them 
  by 
  

   cos 
  mt. 
  Let 
  the 
  superposed 
  coupled 
  vibrations 
  for 
  each 
  

   system 
  be 
  denoted 
  by 
  cos 
  pt 
  and 
  cos 
  qt. 
  Then, 
  for 
  the 
  

   electrical 
  case, 
  we 
  have 
  

  

  p 
  > 
  m 
  > 
  q. 
  

  

  Whereas, 
  for 
  the 
  mechanical 
  analogy, 
  we 
  have 
  

   p 
  = 
  w, 
  and 
  m 
  > 
  q. 
  

  

  5. 
  What 
  would 
  seem 
  to 
  the 
  writers 
  to 
  be 
  an 
  exact 
  

   mechanical 
  analogy 
  to 
  the 
  electrical 
  case 
  would 
  be 
  one 
  

   capable 
  of 
  representation 
  as 
  follows 
  : 
  

  

  t> 
  d*y 
  -„o 
  T 
  d 
  2 
  z 
  ~\ 
  

  

  ?J 
  + 
  ™v 
  = 
  J 
  a* 
  | 
  

  

  k 
  . 
  </ 
  .. 
  . 
  (4) 
  

  

  In 
  these 
  P 
  and 
  Q 
  denote 
  the 
  masses 
  which 
  vibrate 
  in 
  the 
  

   two 
  systems. 
  Their 
  separate 
  vibrations 
  are 
  to 
  be 
  obtained 
  

   by 
  writing 
  J 
  = 
  0. 
  Thus 
  giving 
  as 
  their 
  separate 
  vibrations 
  

   cos 
  mt 
  and 
  cos 
  nt 
  respectively. 
  Further, 
  it 
  should 
  be 
  noted 
  

   that 
  in 
  the 
  above 
  we 
  are 
  supposing 
  that 
  the 
  introduction 
  of 
  

   the 
  cross-connexion 
  terms 
  on 
  the 
  right 
  has 
  not 
  modified 
  

   the 
  coefficients 
  on 
  the 
  left. 
  A 
  model 
  fulfilling 
  these 
  condi- 
  

   tions 
  seems 
  to 
  be 
  still 
  a 
  desideratum. 
  In 
  these 
  equations 
  it 
  

   may 
  be 
  seen 
  by 
  the 
  theory 
  of 
  dimensions 
  that 
  J 
  must 
  be 
  a 
  

   mass 
  like 
  P 
  and 
  Q. 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  35. 
  No. 
  206. 
  Feb. 
  1918. 
  Q 
  

  

  