﻿112 
  Mr. 
  R. 
  Hargreaves 
  

  

  on 
  

  

  system 
  (iii 
  d), 
  and 
  (b) 
  as 
  a 
  function 
  of 
  momenta 
  by 
  eliminat- 
  

   ing 
  <i 
  through 
  (iii 
  c) 
  . 
  I 
  take 
  it 
  to 
  be 
  the 
  criterion 
  of 
  regularity 
  

   in 
  the 
  above 
  system, 
  that 
  it 
  should 
  show 
  the 
  same 
  dynamical 
  

   quantities 
  as 
  the 
  methods 
  of 
  (i) 
  or 
  (ii) 
  when 
  applied 
  to 
  the 
  

   homogeneous 
  forms 
  of 
  E. 
  The 
  requisite 
  analysis 
  is 
  furnished 
  

   by 
  the 
  lemma 
  relating 
  to 
  transformation 
  to 
  be 
  given 
  in 
  the 
  

   next 
  section. 
  This 
  is 
  an 
  extension 
  of 
  the 
  usual 
  theorem 
  of 
  

   reciprocity, 
  and 
  for 
  convenience 
  of 
  general 
  reference 
  it 
  will 
  

   be 
  given 
  in 
  an 
  independent 
  algebraical 
  notation. 
  

  

  § 
  2. 
  Lemma. 
  — 
  Let 
  Z 
  be 
  a 
  quadratic 
  function 
  of 
  n 
  variables 
  

   2, 
  and 
  J 
  a 
  bilinear 
  function 
  of 
  z 
  and 
  of 
  m 
  other 
  variables 
  x, 
  i.e. 
  

  

  m 
  n 
  

  

  J= 
  2 
  X 
  c 
  rs 
  x 
  r 
  z 
  s 
  (2) 
  

  

  r=l 
  s=l 
  

  

  Z 
  can 
  be 
  expressed 
  as 
  a 
  quadratic 
  function 
  of 
  x 
  and 
  of 
  n 
  new 
  

   variables 
  y, 
  specially 
  correlated 
  with 
  z, 
  by 
  means 
  of 
  the 
  n 
  

   linear 
  equations 
  

  

  ^— 
  (say 
  &) 
  =y 
  s 
  + 
  ^ 
  =y 
  s 
  + 
  $c 
  rS 
  xr. 
  ... 
  (3) 
  

   Calling 
  this 
  expression 
  Z(a?,y), 
  we 
  prove 
  

  

  az(s, 
  y) 
  _ 
  a 
  j 
  az(*,y) 
  '_ 
  _ 
  ... 
  

  

  For 
  the 
  first, 
  

  

  "dxr 
  3# 
  r 
  ' 
  By 
  s 
  ts 
  ' 
  ^ 
  ' 
  

  

  ~dZ(x 
  

   ■dx 
  

  

  — 
  ^ 
  V" 
  "N 
  — 
  -n 
  -^ 
  >s~ 
  ^ 
  - 
  s 
  S. 
  

  

  s 
  QX 
  r 
  QX 
  r 
  s 
  s 
  0# 
  r 
  

  

  i.V.^4- 
  ir, 
  

  

  -8^) 
  *** 
  by 
  (3); 
  

  

  that 
  is 
  

  

  For 
  the 
  second 
  

  

  3Z(*,y) 
  _ 
  v 
  ,, 
  „__ 
  dJ 
  

  

  QXr 
  s 
  O-l'r 
  

  

  J 
  

  

  

  ^Z('.r,//) 
  __a_£> 
  _^B6 
  

  

  3y 
  s 
  "dys 
  t 
  i 
  % 
  *Sy 
  s 
  

  

  

  = 
  2 
  BZ 
  ^'' 
  v) 
  c, 
  by 
  (3), 
  

  

  or 
  3Z(. 
  

  

  v) 
  _ 
  , 
  

  

  B/ 
  

  

  '.,• 
  

  

  We 
  then 
  prove 
  tluii 
  if 
  the 
  coefficients 
  in 
  Z 
  and 
  J 
  depend 
  on 
  

  

  any 
  parameter 
  

  

  0, 
  

  

  az(*, 
  y 
  ) 
  a.i 
  ax 
  , 
  m 
  

  

  a# 
  ~a^ 
  a# 
  w 
  

  

  