﻿120 
  Interaction 
  of 
  Dynamical 
  Systems. 
  

  

  whole 
  motion 
  cyclic 
  and 
  acyclic. 
  Momenta 
  and 
  fluxes 
  are 
  

   then 
  derived 
  from 
  a 
  kinetic 
  potential 
  of 
  mixed 
  type, 
  viz. 
  : 
  — 
  

  

  L=T-K+I 
  

  

  , 
  BL 
  ; 
  

  

  in 
  precise 
  correspondence 
  with 
  the 
  general 
  scheme. 
  

  

  In 
  the 
  above 
  statement 
  of 
  the 
  case, 
  the 
  evidence 
  of 
  inter- 
  

   action 
  is 
  supposed 
  to 
  be 
  first 
  noticed 
  in 
  connexion 
  with 
  the 
  

   value 
  of 
  momentum 
  : 
  the 
  modification 
  of 
  flux 
  obtained 
  by 
  

   assuming 
  the 
  regularity 
  of 
  the 
  dynamical 
  system 
  then 
  takes 
  

   an 
  entirely 
  convincing 
  form. 
  If 
  the 
  modifications 
  are 
  taken 
  

   concurrently, 
  the 
  proof 
  of 
  regularity 
  is 
  gis^en 
  by 
  the 
  co- 
  

   existence 
  of 
  

  

  , 
  dT 
  BI 
  • 
  3K 
  BI 
  

  

  + 
  7w 
  and 
  % 
  = 
  ^T-^' 
  

  

  "du 
  Bw' 
  * 
  "dx 
  "d 
  

  

  K 
  

  

  I 
  being 
  a 
  bilinear 
  function 
  of 
  u 
  and 
  k. 
  The 
  whole 
  forms 
  an 
  

   excellent 
  example 
  of 
  the 
  mode 
  in 
  which 
  the 
  coefficients 
  of 
  I. 
  

   are 
  determined 
  in 
  a 
  special 
  problem. 
  As 
  the 
  expression 
  for 
  

   I 
  in 
  (26) 
  is 
  an 
  interpretable 
  quantity, 
  its 
  treatment 
  in 
  the 
  

   further 
  development 
  for 
  several 
  bodies 
  and 
  general 
  coordinates 
  

   offers 
  no 
  difficulties 
  which 
  do 
  not 
  also 
  appear 
  in 
  treating 
  

   T 
  and 
  K. 
  

  

  The 
  conception 
  of 
  correlated 
  expressions 
  for 
  energy 
  is 
  

   not 
  confined 
  to 
  mechanics, 
  and 
  the 
  generalized 
  theorem 
  of 
  

   reciprocity 
  in 
  the 
  lemma 
  is 
  applicable 
  in 
  all 
  cases. 
  Maxwell 
  

   gave 
  two 
  such 
  expressions 
  for 
  the 
  energy 
  of 
  a 
  system 
  of 
  

   insulated 
  conductors, 
  one 
  a 
  quadratic 
  function 
  of 
  charges, 
  the 
  

   other 
  a 
  quadratic 
  function 
  of 
  potentials. 
  The 
  former 
  has 
  a 
  

   close 
  analogy 
  with 
  the 
  function 
  of 
  constants 
  of 
  circulation 
  

   appearing 
  in 
  Kelvin's 
  problem. 
  In 
  each 
  case 
  we 
  have 
  in 
  the 
  

   character 
  of 
  momentum 
  a 
  quantity 
  unalterable 
  so 
  long 
  as 
  the 
  

   degrees 
  of 
  freedom 
  of 
  the 
  system 
  are 
  maintained, 
  a 
  quantity 
  

   which 
  is 
  virtually 
  a 
  constant 
  of 
  integration 
  in 
  one 
  group 
  of 
  

   the 
  equations. 
  It 
  may 
  be 
  well 
  to 
  bear 
  in 
  mind, 
  in 
  dealing 
  

   with 
  different 
  branches 
  of 
  physics, 
  that 
  the 
  existence 
  of 
  a 
  

   total 
  energy 
  in 
  the 
  form 
  of 
  detached 
  expressions 
  docs 
  not 
  

   exclude 
  interaction 
  between 
  the 
  separate 
  forms 
  of 
  energy. 
  

  

  