﻿118 
  Mr. 
  R. 
  Hargrcavcs 
  on 
  

  

  in 
  a 
  general 
  way 
  by 
  the 
  use 
  of 
  a 
  mixed 
  kinetic 
  potential, 
  

  

  L 
  = 
  QOJ)-PW+I(^). 
  . 
  . 
  . 
  (iii&) 
  

  

  On 
  the 
  other 
  hand, 
  L 
  may 
  appear 
  as 
  the 
  difference 
  of 
  two 
  

   detached 
  energy-forms 
  of 
  like 
  type, 
  viz. 
  : 
  — 
  

  

  L 
  = 
  Q(<7)-Q'(*) 
  (iva) 
  

  

  or 
  L 
  = 
  PG?)-P(w) 
  (vo) 
  

  

  This 
  is 
  consistent 
  with 
  a 
  mutual 
  action 
  expressed 
  by 
  the 
  

   use 
  of 
  

  

  E 
  = 
  Q(£) 
  + 
  Q'(%H-iar 
  ) 
  . 
  . 
  . 
  (ivb) 
  

   or 
  B«Pfr)+F(*r)+I(p,«0 
  .... 
  (t6) 
  

  

  respectively, 
  as 
  energy-forms 
  and 
  at 
  the 
  same 
  time 
  kinetic 
  

   potentials 
  of 
  types 
  (i) 
  and 
  (ii). 
  But 
  here 
  I 
  must 
  be 
  such 
  as 
  

   to 
  make 
  E 
  essentially 
  positive. 
  

  

  In 
  relation 
  to 
  the 
  product-forms 
  j2jo» 
  ( 
  h 
  aR 
  d 
  i^^X*? 
  E 
  is 
  

  

  r 
  s 
  

  

  in 
  all 
  cases 
  the 
  sum, 
  and 
  L 
  the 
  difference. 
  E 
  is 
  a 
  kinetie 
  

   potential 
  when 
  its 
  expression 
  is 
  homogeneous, 
  L 
  is 
  a 
  kinetic 
  

   potential 
  when 
  its 
  expression 
  is 
  heterogeneous. 
  When 
  E 
  

   is 
  expressed 
  as 
  a 
  sum 
  of 
  unlike 
  forms, 
  or 
  L 
  as 
  a 
  difference 
  of 
  

   like 
  forms, 
  all 
  mutual 
  action 
  of 
  the 
  groups 
  or 
  systems 
  is 
  

   concealed, 
  and 
  these 
  forms 
  are 
  unsuited 
  for 
  use 
  as 
  kinetic 
  

   potentials. 
  

  

  § 
  9. 
  As 
  an 
  example 
  of 
  the 
  method 
  we 
  take 
  the 
  problem 
  of 
  

   a 
  perforated 
  solid 
  moving 
  in 
  a 
  perfect 
  liquid. 
  Kirchoff 
  

   wrote 
  a 
  potential 
  ^r 
  of 
  acyclic 
  motion 
  in 
  the 
  form 
  

  

  and 
  obtained 
  an 
  energy 
  function 
  T 
  quadratic 
  in 
  ?/...?■. 
  Here 
  

   (uvw) 
  is 
  translation, 
  (pqr) 
  rotation 
  ; 
  in 
  some 
  respects 
  it 
  is 
  

   more 
  convenient 
  to 
  use 
  Wj...w 
  6 
  , 
  and 
  then 
  to 
  write 
  fi...& 
  for 
  

  

  tvi 
  

  

  components 
  of 
  momentum 
  linear 
  and 
  angular, 
  i. 
  e. 
  f 
  or 
  — 
  ,. 
  . 
  .. 
  

  

  Kelvin 
  added 
  a 
  cyclic 
  potential 
  ft>=tf 
  1 
  &> 
  1 
  -ftf 
  2 
  a> 
  2 
  + 
  ..., 
  and 
  

   obtained 
  an 
  energy-function 
  K 
  quadratic 
  in 
  k. 
  The 
  energy 
  

   of 
  the 
  whole 
  motion 
  is 
  T 
  + 
  K, 
  involving 
  no 
  products 
  of 
  u 
  and 
  *, 
  

   and 
  there 
  are 
  reasons 
  for 
  assigning 
  to 
  k 
  the 
  character 
  of 
  a 
  

   momentum. 
  Now 
  when 
  the 
  momenta 
  of 
  the 
  liquid 
  motion 
  

   are 
  found 
  by 
  evaluating 
  

  

  j]>" 
  da 
  dl) 
  (h, 
  jQJp 
  {.rr- 
  t/ll 
  )d,r 
  dijdz..., 
  

  

  