﻿114 
  Mr. 
  R. 
  Har 
  greaves 
  on 
  

  

  function 
  of 
  q 
  and 
  ot 
  having 
  the 
  part 
  quadratic 
  in 
  q 
  on 
  ly 
  

   essentially 
  positive, 
  that 
  in 
  ot 
  only 
  essentially 
  negative 
  ; 
  

   these 
  conditions 
  secure 
  the 
  positive 
  character 
  of 
  E. 
  The 
  

   latter 
  is 
  derived 
  from 
  L 
  by 
  the 
  formulas 
  

  

  E=2* 
  r 
  |£ 
  _L, 
  orE 
  = 
  2^^ 
  L) 
  -(-L) 
  = 
  L-^ 
  s 
  | 
  L 
  ^j 
  

  

  r 
  Mr 
  0™s 
  s 
  O^s 
  1 
  /gN 
  

  

  and 
  2E=2&H 
  -^1^. 
  ! 
  

  

  The 
  alternative 
  transformation 
  is 
  to 
  make 
  use 
  of 
  (iii 
  c) 
  to 
  

   eliminate 
  q 
  and 
  express 
  E 
  as 
  E^-sr) 
  ; 
  as 
  the 
  latter 
  is 
  the 
  

   reciprocal 
  of 
  E(^,^) 
  it 
  is 
  not 
  necessary 
  to 
  give 
  details. 
  It 
  

   is 
  also 
  possible 
  to 
  use 
  (iii 
  c) 
  and 
  (iii 
  d) 
  in 
  conjunction 
  to 
  effect 
  

   a 
  transformation 
  from 
  q 
  and 
  ot 
  to 
  p 
  and 
  & 
  ? 
  these 
  being 
  

   m 
  + 
  n 
  linear 
  equations. 
  It 
  will 
  be 
  sufficient 
  to 
  state 
  that 
  

   ~L(p,X) 
  is 
  the 
  reciprocal 
  of 
  L($,ot), 
  i. 
  e. 
  L(p,#) 
  is 
  a 
  kinetic 
  

   potential 
  of 
  type 
  (ii) 
  for 
  the 
  variables 
  q, 
  and 
  — 
  L(j9, 
  &) 
  is 
  a 
  

   kinetic 
  potential 
  of 
  type 
  (i) 
  for 
  the 
  variables 
  %. 
  Thus 
  

  

  . 
  BL( 
  P 
  ,X) 
  3LQ>,*) 
  . 
  n 
  ,9L(p,X) 
  3L(j,w)_j 
  

  

  9 
  ' 
  = 
  ^^~ 
  ' 
  OTs= 
  §*T 
  ' 
  ~^~ 
  ~~^ 
  " 
  

  

  is 
  true 
  for 
  coordinates 
  of 
  either 
  type. 
  

  

  § 
  4. 
  The 
  force 
  corresponding 
  to 
  the 
  coordinate 
  q 
  r 
  is, 
  by 
  (iii 
  e), 
  

  

  ^ 
  d£r 
  d?r 
  3?r 
  dt 
  Mr 
  Mr 
  ^ 
  ' 
  

  

  If 
  I 
  is 
  taken 
  to 
  be 
  

  

  %c 
  rs 
  q 
  r 
  *r 
  s 
  , 
  (8) 
  

  

  r,s 
  

  

  the 
  terms 
  due 
  to 
  the 
  function 
  of 
  interaction 
  are 
  

  

  d 
  31 
  _ 
  31 
  

  

  dt 
  Mr 
  Mr 
  

  

  ■ 
  -?^ 
  + 
  *fe-*-" 
  + 
  l(fer-^->-- 
  (9) 
  

  

  The 
  force 
  corresponding 
  to 
  the 
  coordinate 
  j£, 
  is 
  

   3P_BQ 
  31 
  

  

  If 
  no 
  coordinates 
  % 
  appear 
  in 
  (lie 
  coefficients 
  of 
  PQI, 
  and 
  

  

  