﻿116 
  Mr. 
  R. 
  Hargr 
  eaves 
  on 
  

  

  heterogeneous 
  expression 
  for 
  energy. 
  Here 
  we 
  suppose 
  E 
  

   given 
  as 
  

  

  Qn(y) 
  + 
  Qi 
  2 
  (?,x) 
  + 
  Q22(x)» 
  • 
  • 
  • 
  • 
  ( 
  15 
  ) 
  

  

  and 
  that 
  % 
  is 
  to 
  be 
  replaced 
  by 
  ta, 
  through 
  the 
  equation 
  

  

  BE 
  dQi2 
  , 
  BQ22 
  , 
  BQ22 
  /im 
  

  

  OTs= 
  §*, 
  - 
  -w 
  + 
  w 
  =Ks+ 
  ~m; 
  sa 
  - 
  v> 
  • 
  (16) 
  

  

  Q 
  12 
  being 
  the 
  bilinear 
  section 
  so 
  that 
  tc 
  s 
  ' 
  j 
  a 
  linear 
  function 
  

   o£ 
  q. 
  The 
  actual 
  transformation 
  needs 
  minor 
  lemma. 
  If 
  

   Q(V) 
  is 
  a 
  quadratic 
  function 
  of 
  x, 
  and 
  lv(j/) 
  its 
  reciprocal, 
  i.e. 
  

   the 
  function 
  obtained 
  by 
  transforming 
  to 
  variables 
  ?/, 
  where 
  

  

  y 
  s 
  = 
  ^— 
  5 
  then 
  the 
  transformation 
  o/StfA 
  + 
  QW 
  % 
  means 
  °f 
  

  

  y 
  s 
  =fc 
  s 
  + 
  ^— 
  , 
  0?' 
  i£s 
  equivalent 
  x 
  s 
  = 
  ^~ 
  K.(y 
  — 
  fc) 
  . 
  (17) 
  

   makes 
  %fc 
  s 
  % 
  s 
  + 
  Q(#) 
  = 
  K(y) 
  — 
  K(k). 
  . 
  (18) 
  

  

  « 
  oy* 
  s 
  L 
  oy 
  s 
  OKs 
  J 
  

  

  =2« 
  S 
  ^-2K(«) 
  . 
  (19 
  a) 
  

  

  and 
  Q(. 
  S 
  )=K(y-«) 
  = 
  K(y) 
  + 
  K(*)-2«^JW; 
  . 
  (19 
  J) 
  

  

  s 
  (J<Js 
  

  

  and 
  (18) 
  follows 
  by 
  addition. 
  

  

  Using 
  K 
  now 
  for 
  the 
  reciprocal 
  of 
  Q 
  22 
  , 
  we 
  have 
  

  

  Bfl,«) 
  = 
  {fti(4)-KWJ+KW=T(J) 
  + 
  K(«) 
  say. 
  (20) 
  

   At 
  the 
  same 
  time 
  L, 
  defined 
  as 
  ^\p 
  r 
  q 
  r 
  — 
  Sj«r^, 
  is 
  equal 
  to 
  

  

  r 
  5 
  

  

  Qn(# 
  ) 
  "" 
  Q22W 
  5 
  an( 
  ^ 
  this 
  expressed 
  in 
  terms 
  of 
  q^r 
  is 
  

   L(4,-)»Qatt)-K(wr«) 
  

  

  -T(J)-KW+^. 
  . 
  . 
  . 
  (21) 
  

  

  Thus 
  in 
  (20) 
  we 
  do 
  in 
  fact 
  reach 
  the 
  energy-form 
  treated 
  

   in 
  § 
  1, 
  and 
  L((/,ot) 
  is 
  the 
  difference 
  of 
  the 
  quadratics 
  in 
  (20) 
  

   with 
  a 
  bilinear 
  function 
  of 
  ([ 
  and 
  m 
  added. 
  

  

  § 
  7. 
  The 
  mutual 
  action 
  of 
  the 
  two 
  systems 
  in 
  § 
  1 
  appears 
  

   in 
  the 
  facts 
  that 
  p 
  r 
  contains 
  terms 
  linear 
  in 
  or, 
  andy 
  contains 
  

   terms 
  linear 
  in 
  <). 
  The 
  coefficients 
  of 
  either 
  set 
  of 
  terms 
  are 
  

   the 
  coefficients 
  in 
  the 
  function 
  of 
  interaction, 
  which 
  there- 
  

   fore 
  expresses 
  the 
  mutual 
  action 
  directly. 
  Suppose 
  the 
  

  

  