﻿450 
  

  

  Prof. 
  Cayley 
  on 
  Prepotentials. 
  

  

  [June 
  10, 
  

  

  and 
  consequently 
  by 
  the 
  integral 
  

  

  r 
  od.v... 
  ch 
  (A) 
  

  

  J{(a-.r)\..+(c-z)* 
  + 
  ^ 
  

  

  that 
  is 
  by 
  the 
  prepotential-plane 
  integral 
  ; 
  but 
  the 
  equation 
  is 
  not 
  

   satisfied 
  by 
  the 
  value 
  

  

  1 
  

  

  {(a-xf.. 
  +( 
  c 
  -zy 
  + 
  (e-ivy} 
  

  

  nor, 
  therefore, 
  by 
  the 
  prepotential-solid, 
  or 
  general 
  superficial, 
  integral. 
  

   But 
  i£ 
  q— 
  — 
  j, 
  then, 
  instead 
  of 
  the 
  prepotential 
  equation, 
  we 
  have 
  

  

  and 
  this 
  is 
  satisfied 
  by 
  

  

  and 
  therefore 
  also 
  by 
  

  

  ,« 
  3 
  .. 
  +c 
  a 
  + 
  e 
  2 
  } 
  

  

  1 
  

  

  { 
  (a 
  - 
  ca) 
  2 
  ...+( 
  C 
  -zy 
  + 
  (e- 
  wy}*-* 
  

   Hence 
  it 
  is 
  satisfied 
  by 
  

  

  p 
  dec 
  . 
  . 
  .dz 
  dw 
  

  

  Ji 
  

  

  the 
  potential-solid 
  integral, 
  provided 
  that 
  the 
  point 
  (a 
  . 
  . 
  . 
  c, 
  <?) 
  c?oes 
  ^io^ 
  lie 
  

   within 
  the 
  material 
  space 
  : 
  I 
  would 
  rather 
  say 
  that 
  the 
  integral 
  does 
  not 
  

   satisfy 
  the 
  equation, 
  but 
  of 
  this 
  more 
  hereafter; 
  and 
  it 
  is 
  satisfied 
  by 
  

  

  J' 
  

  

  ^ 
  . 
  .. 
  . 
  . 
  (0) 
  

  

  {(a-xy 
  . 
  . 
  . 
  +(c-zy 
  + 
  (e-wy} 
  

  

  the 
  potential-surface 
  integral. 
  The 
  potential-plane 
  integral 
  (B), 
  as 
  a 
  

   particular 
  case 
  of 
  (C), 
  of 
  course 
  also 
  satisfies 
  the 
  equation. 
  

  

  Each 
  of 
  the 
  four 
  cases 
  gives 
  rise 
  to 
  what 
  may 
  be 
  called 
  a 
  distribution- 
  

   theorem, 
  viz. 
  given 
  Y 
  a 
  function 
  of 
  (a 
  . 
  . 
  . 
  c, 
  e) 
  satisfying 
  certain 
  prescribed 
  

   conditions, 
  but 
  otherwise 
  arbitrary, 
  then 
  the 
  form 
  of 
  the 
  theorem 
  is 
  that 
  

   there 
  exists 
  and 
  can 
  be 
  found 
  an 
  expression 
  for 
  p, 
  the 
  density 
  or 
  distri- 
  

   bution 
  of 
  matter 
  over 
  the 
  space 
  or 
  surface 
  to 
  which 
  the 
  theorem 
  relates, 
  

   such 
  that 
  the 
  corresponding 
  integral 
  V 
  has 
  its 
  given 
  value, 
  viz. 
  that 
  in 
  A 
  

   and 
  B 
  there 
  exists 
  such 
  a 
  distribution 
  over 
  the 
  plane 
  «' 
  = 
  0, 
  in 
  C 
  such 
  a 
  

   distribution 
  over 
  a 
  given 
  surface, 
  and 
  in 
  D 
  such 
  a 
  distribution 
  in 
  space. 
  

   The 
  establishment, 
  and 
  exhibition 
  in 
  connexion 
  with 
  each 
  other, 
  of 
  these 
  

  

  