﻿Interaction 
  of 
  Dynamical 
  Systems. 
  113 
  

  

  where 
  in 
  Z(x,y) 
  the 
  coefficients 
  of 
  x 
  2 
  , 
  xy 
  . 
  , 
  . 
  are 
  differentiated, 
  

   in 
  Z 
  the 
  coefficients 
  of 
  z^..., 
  and 
  in 
  J 
  those 
  of 
  x 
  r 
  z 
  s 
  . 
  Thus 
  

   2i(x,y) 
  depends 
  on 
  6 
  directly 
  through 
  the 
  coefficients 
  of 
  Z(V), 
  

   and 
  in 
  an 
  implicit 
  way 
  because 
  z 
  as 
  expressed 
  in 
  terms 
  of 
  xy 
  

   is 
  also 
  a 
  function 
  of 
  6 
  through 
  the 
  coefficients. 
  Hence 
  

  

  az(. 
  v/ 
  ) 
  _az 
  <r 
  a*._3z 
  a 
  gf 
  . 
  rra? 
  

  

  -3T 
  +2 
  ~a0 
  T^S?' 
  by 
  W 
  " 
  

  

  BJ 
  

  

  But 
  ^-g 
  is 
  a 
  linear 
  function 
  of 
  

   Of 
  

  

  written 
  for 
  <v 
  s 
  , 
  and 
  therefore 
  

  

  viz., 
  it 
  is 
  J 
  with 
  

  

  

  X*. 
  

  

  & 
  s 
  -de 
  

  

  ^ 
  and 
  so 
  h$ 
  +^-^0. 
  

  

  In 
  the 
  usual 
  theorem 
  of 
  reciprocity, 
  the 
  letters 
  x 
  and 
  the 
  

   function 
  J 
  do 
  not 
  occur. 
  

  

  § 
  3. 
  To 
  express 
  E 
  or 
  Q(#)-f-P(V) 
  in 
  terms 
  of 
  velocities 
  

   only, 
  we 
  make 
  use 
  of 
  

  

  5P 
  _ 
  i 
  31 
  

  

  S 
  A-S+ 
  ^ 
  

  

  (i. 
  e. 
  iii 
  rf), 
  

  

  to 
  eliminate 
  «r 
  ; 
  consequently 
  (ZJzyx) 
  of 
  the 
  lemma 
  are 
  re- 
  

   placed 
  by 
  (PIw^; 
  £). 
  Hence 
  

  

  by 
  (A) 
  Sm*)=|L, 
  » 
  = 
  J 
  

  

  ' 
  *■ 
  ' 
  Sir 
  &* 
  2>* 
  s 
  ' 
  I 
  

  

  BP(<Z,*)_dI_9P 
  

   ~~p 
  30 
  a*' 
  

  

  and 
  by 
  (B) 
  

  

  8 
  being 
  one 
  of 
  the 
  coordinates 
  q 
  or 
  %, 
  or 
  indeed 
  any 
  other 
  

   parameter 
  involved 
  in 
  the 
  coefficients 
  of 
  P 
  Q 
  and 
  I. 
  Thus 
  

  

  BE(g,x) 
  

  

  90. 
  

   3& 
  

  

  + 
  w.=«r 
  B 
  

  

  and 
  

  

  3E( 
  j,tf) 
  _ 
  BQ 
  . 
  BI 
  _ 
  c*P_ 
  3L 
  

   — 
  -33— 
  ~ 
  ~66 
  + 
  B0 
  B# 
  B# 
  * 
  

  

  Thus 
  the 
  criterion, 
  stated 
  in 
  § 
  1, 
  for 
  the 
  normal 
  character 
  

   of 
  the 
  system 
  derived 
  from 
  the 
  use 
  of 
  the 
  mixed 
  kinetic 
  

   potential 
  L, 
  is 
  satisfied. 
  Here 
  L 
  is 
  a 
  general 
  quadratic 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  16. 
  No. 
  91. 
  July 
  1908. 
  I 
  

  

  K5) 
  

  

  