﻿Pressure-integral 
  as 
  Kinetic 
  Potential. 
  441 
  

  

  These 
  values 
  make 
  L 
  equal 
  to 
  0^-f 
  2 
  e>+ 
  ... 
  (or 
  X 
  with 
  

   more 
  variables), 
  i. 
  e. 
  the 
  value 
  of 
  E 
  found 
  by 
  the 
  process 
  (9) 
  

   is 
  zero, 
  and 
  also 
  Ye 
  or 
  — 
  O 
  is 
  zero. 
  [Or 
  we 
  may 
  show 
  that 
  

  

  ^ 
  + 
  1)— 
  (7i), 
  (r) 
  — 
  (r-1) 
  + 
  n 
  (r) 
  5 
  ^/3 
  ~ 
  dt 
  Tsd' 
  ' 
  

  

  where 
  the 
  middle 
  form 
  applies 
  in 
  all 
  cases 
  but 
  the 
  first 
  and 
  

   last. 
  The 
  construction 
  of 
  @'s 
  by 
  (8) 
  or 
  the 
  footnote 
  leads 
  

   then 
  to 
  the 
  above 
  result.] 
  

  

  Thus 
  if 
  L 
  is 
  a 
  kinetic 
  potential 
  of 
  any 
  order 
  a 
  function 
  E 
  

   is 
  derivable 
  from 
  it 
  by 
  the 
  direct 
  process 
  in 
  (9), 
  which 
  will 
  

   have 
  the 
  character 
  of 
  energy, 
  i. 
  e. 
  its 
  rate 
  of 
  increment 
  may 
  

  

  be 
  expressed 
  as 
  a 
  rate 
  of 
  working 
  of 
  forces 
  F. 
  But 
  if 
  L= 
  -y> 
  

  

  f 
  defined 
  as 
  above, 
  then 
  the 
  forces 
  concerned 
  and 
  the 
  value 
  

   of 
  E 
  derived 
  from 
  L 
  all 
  vanish. 
  

  

  The 
  term 
  then 
  by 
  which 
  P 
  differs 
  from 
  Kelvin's 
  kinetic 
  

   potential 
  contributes 
  nothing 
  to 
  force 
  or 
  to 
  the 
  expression 
  

   for 
  energy. 
  As 
  applied 
  to 
  L 
  the 
  formula 
  * 
  gives 
  

  

  E 
  = 
  2£— 
  L 
  -L 
  = 
  2T 
  + 
  I-(T-K 
  + 
  I) 
  = 
  T 
  + 
  K, 
  

  

  since 
  I 
  is 
  a 
  linear 
  function 
  of 
  #, 
  and 
  K 
  does 
  not 
  contain 
  6. 
  

   Therefore 
  P 
  and 
  L 
  when 
  used 
  as 
  kinetic 
  potentials 
  yield 
  the 
  

   same 
  forces, 
  and 
  a 
  correct 
  value 
  of 
  the 
  energy. 
  

  

  A 
  minor 
  example 
  of 
  the 
  principle 
  of 
  this 
  section 
  is 
  the 
  

   equivalence 
  of 
  ir?/, 
  —.vy, 
  and 
  \(ooy 
  — 
  xif) 
  as 
  kinetic 
  potentials, 
  

   forms 
  occurring 
  in 
  different 
  estimates 
  of 
  I 
  for 
  a 
  circular 
  

   cylinder. 
  The 
  case 
  where 
  / 
  is 
  linear 
  in 
  velocities, 
  and 
  so 
  

   1-F 
  

   -TT 
  has 
  a 
  part 
  linear 
  in 
  accelerations 
  and 
  another 
  part 
  quadratic 
  

  

  in 
  velocities, 
  seems 
  likely 
  to 
  be 
  of 
  common 
  occurrence. 
  

  

  An 
  important 
  example 
  is 
  the 
  electromagnetic 
  formula 
  for 
  

   volume 
  distribution 
  given 
  in 
  Lorentz, 
  Encykl. 
  Math. 
  Wiss. 
  

   v. 
  ii. 
  p. 
  160, 
  viz. 
  

  

  j*(i2a»-i2X^T= 
  Ji(2F«/V--f 
  )prfr- 
  ~ 
  | 
  J2XF 
  

  

  An 
  example 
  of 
  both 
  sections 
  is 
  furnished 
  by 
  Green's 
  

  

  * 
  We 
  have 
  also 
  E 
  = 
  L 
  -2k~ 
  . 
  On 
  the 
  use 
  of 
  a 
  kinetic 
  potential 
  of 
  

   this 
  type, 
  and 
  the 
  coefficients 
  in 
  I, 
  compare 
  Phil. 
  Mag. 
  July 
  1908. 
  

  

  