﻿1875.] 
  Prof. 
  Cayley 
  on 
  Prepotentials. 
  449 
  

  

  stated, 
  assumed 
  that 
  the 
  attracted 
  point 
  does 
  not 
  lie 
  on 
  the 
  material 
  sur- 
  

   face 
  ; 
  to 
  make 
  it 
  do 
  so 
  is, 
  in 
  fact, 
  a 
  particular 
  supposition. 
  As 
  to 
  solid 
  

   integrals, 
  the 
  cases 
  where 
  the 
  attracted 
  point 
  is 
  not, 
  and 
  is, 
  in 
  the 
  material 
  

   space 
  may 
  be 
  regarded 
  as 
  cases 
  of 
  coordinate 
  generality; 
  or 
  we 
  may 
  

   regard 
  the 
  latter 
  one 
  as 
  the 
  general 
  case, 
  deducing 
  the 
  former 
  one 
  from 
  

   it 
  by 
  supposing 
  the 
  density 
  at 
  the 
  attracted 
  point 
  to 
  become 
  =0. 
  

  

  The 
  present 
  memoir 
  has 
  chiefly 
  reference 
  to 
  three 
  principal 
  cases, 
  

   which 
  I 
  call 
  A, 
  C, 
  D, 
  and 
  a 
  special 
  case, 
  B, 
  included 
  both 
  under 
  A 
  and 
  

   C 
  : 
  viz. 
  these 
  are 
  : 
  — 
  

  

  A. 
  The 
  prepotential-plane 
  case 
  ; 
  q 
  general, 
  but 
  the 
  surface 
  is 
  here 
  the 
  

   plane 
  iv 
  = 
  0, 
  so 
  that 
  the 
  integral 
  is 
  

  

  dec 
  . 
  . 
  . 
  dz 
  

  

  B. 
  The 
  potential-plane 
  case 
  ; 
  q= 
  — 
  i, 
  and 
  the 
  surface 
  the 
  plane 
  if 
  =0, 
  

   so 
  that 
  the 
  integral 
  is 
  

  

  p 
  dec 
  . 
  . 
  . 
  dz 
  

  

  C. 
  The 
  potential-surface 
  case; 
  q 
  = 
  — 
  ^, 
  the 
  surface 
  arbitrary, 
  so 
  that 
  

   the 
  integral 
  is 
  

  

  dS 
  

  

  {(a-ivf 
  . 
  . 
  . 
  + 
  (c-zf 
  + 
  (e 
  - 
  wf}l 
  

   D. 
  The 
  potential-solid 
  case 
  ; 
  q= 
  — 
  |, 
  and 
  the 
  integral 
  is 
  

  

  p 
  doe 
  . 
  . 
  . 
  dz 
  dw 
  

  

  {{a 
  - 
  . 
  . 
  . 
  + 
  (c 
  - 
  zf 
  + 
  (e-iu)Y 
  s 
  ~ 
  * 
  

  

  It 
  is, 
  in 
  fact, 
  only 
  the 
  prepotential-plane 
  case 
  which 
  is 
  connected 
  with 
  

   the 
  partial 
  differential 
  equation 
  

  

  (f 
  2 
  ..+ 
  £ 
  + 
  £ 
  + 
  ?£±iAv=o, 
  

  

  \acr 
  dcr 
  de 
  e 
  de 
  / 
  

  

  considered 
  in 
  Green's 
  memoir 
  £ 
  On 
  the 
  Attractions 
  of 
  Ellipsoids 
  ' 
  (1835), 
  

   and 
  called 
  here 
  the 
  prepotential 
  equation. 
  Eor 
  this 
  equation 
  is 
  satis- 
  

   fied 
  by 
  the 
  function 
  

  

  1 
  

  

  {a 
  2 
  ... 
  +c 
  2 
  + 
  <r} 
  i6 
  ' 
  +2 
  ' 
  

  

  and 
  therefore 
  also 
  by 
  

  

  2n2 
  

  

  