﻿448 
  

  

  Prof. 
  Cayley 
  on 
  Prepotentials. 
  [June 
  10_, 
  

  

  point 
  (x 
  . 
  . 
  z, 
  w\ 
  the 
  corresponding 
  density 
  is 
  p, 
  a 
  given 
  function 
  of 
  

   (x 
  . 
  . 
  . 
  z, 
  iv), 
  and 
  that 
  the 
  element 
  of 
  mass 
  pd™ 
  exerts 
  on 
  the 
  attracted 
  

   point 
  (cc.c, 
  e) 
  a 
  force 
  proportional 
  to 
  the 
  (s 
  + 
  2q+l)th 
  power 
  of 
  the 
  

   distance 
  {(a—x) 
  2 
  .. 
  +(c 
  — 
  z) 
  2 
  + 
  (<?— 
  iu) 
  2 
  )^. 
  The 
  integration 
  is 
  extended 
  

   so 
  as 
  to 
  include 
  the 
  whole 
  attracting 
  mass 
  Jpd^uy 
  ; 
  and 
  the 
  integral 
  is 
  then 
  

   said 
  to 
  represent 
  the 
  prepotential 
  of 
  the 
  mass 
  in 
  regard 
  to 
  the 
  point 
  

   (a 
  . 
  . 
  c, 
  e). 
  In 
  the 
  particular 
  case 
  s=2, 
  q— 
  — 
  j, 
  the 
  force 
  is 
  as 
  the 
  inverse 
  

   square 
  of 
  the 
  distance, 
  and 
  the 
  integral 
  represents 
  the 
  potential 
  in 
  the 
  

   ordinary 
  sense 
  of 
  the 
  word. 
  

  

  The 
  element 
  of 
  volume 
  d<& 
  is 
  usually 
  either 
  the 
  element 
  of 
  solid 
  

   (spatial 
  or 
  (s 
  + 
  i 
  )-dii:nensional) 
  volume 
  dx 
  . 
  . 
  . 
  dzdiv, 
  or 
  else 
  the 
  element 
  

   of 
  superficial 
  (s-dimensional) 
  volume 
  d$. 
  In 
  particular, 
  when 
  the 
  surface 
  

   (s-dimensional 
  locus) 
  is* 
  the 
  (s-dimensional) 
  plane 
  iv 
  = 
  0, 
  the 
  superficial 
  

   element 
  dS 
  is=dx 
  . 
  . 
  . 
  dz. 
  The 
  cases 
  of 
  a 
  less-than-s-dimensional 
  volume 
  

   are 
  in 
  the 
  present 
  memoir 
  considered 
  only 
  incidentally. 
  It 
  is 
  scarcely 
  

   necessary 
  to 
  remark 
  that 
  the 
  notion 
  of 
  density 
  is 
  dependent 
  on 
  the 
  dimen- 
  

   sionality 
  of 
  the 
  element 
  of 
  volume 
  d™ 
  : 
  in 
  passing 
  from 
  a 
  spatial 
  dis- 
  

   tribution, 
  p 
  dx 
  . 
  . 
  . 
  dzdiv, 
  to 
  a 
  superficial 
  distribution, 
  p 
  dS, 
  we 
  alter 
  the 
  

   signification 
  of 
  p. 
  In 
  fact 
  if, 
  in 
  order 
  to 
  connect 
  the 
  two, 
  we 
  imagine 
  

   the 
  spatial 
  distribution 
  as 
  made 
  over 
  an 
  indefinitely 
  thin 
  layer 
  or 
  stratum 
  

   bounded 
  by 
  the 
  surface, 
  so 
  that 
  at 
  any 
  element 
  dS 
  of 
  the 
  surface 
  the 
  

   normal 
  thickness 
  is 
  dp, 
  where 
  dv 
  is 
  a 
  function 
  of 
  the 
  coordinates 
  (x 
  . 
  . 
  z, 
  w) 
  

   of 
  the 
  element 
  dS, 
  the 
  spatial 
  element 
  is 
  —dv 
  c?S, 
  and 
  the 
  element 
  of 
  mass 
  

   p 
  dx. 
  .dzdw 
  is 
  =pdvdS 
  ; 
  and 
  then 
  changing 
  the 
  signification 
  of 
  p, 
  so 
  as 
  

   to 
  denote 
  by 
  it 
  the 
  product 
  p 
  dv, 
  the 
  expression 
  for 
  the 
  element 
  of 
  mass 
  

   becomes 
  pdS, 
  which 
  is 
  the 
  formula 
  in 
  the 
  case 
  of 
  the 
  superficial 
  dis- 
  

   tribution. 
  

  

  The 
  space 
  or 
  surface 
  over 
  which 
  the 
  distribution 
  extends 
  may 
  be 
  spoken 
  

   of 
  as 
  the 
  material 
  space 
  or 
  surface 
  ; 
  so 
  that 
  the 
  density 
  p 
  is 
  not 
  = 
  for 
  

   any 
  finite 
  portion 
  of 
  the 
  material 
  space 
  or 
  surface 
  ; 
  and 
  if 
  the 
  distribution 
  

   be 
  such 
  that 
  the 
  density 
  becomes 
  = 
  for 
  any 
  point 
  or 
  locus 
  of 
  the 
  mate- 
  

   rial 
  space 
  or 
  surface, 
  then 
  such 
  point 
  or 
  locus, 
  considered 
  as 
  an 
  infini- 
  

   tesimal 
  portion 
  of 
  space 
  or 
  surface, 
  may 
  be 
  excluded 
  from 
  and 
  regarded 
  

   as 
  not 
  belonging 
  to 
  the 
  material 
  space 
  or 
  surface. 
  It 
  is 
  allowable, 
  and 
  

   frequently 
  convenient, 
  to 
  regard 
  p 
  as 
  a 
  discontinuous 
  function, 
  having 
  its 
  

   proper 
  value 
  within 
  the 
  material 
  space 
  or 
  surface, 
  and 
  having 
  else- 
  

   where 
  the 
  value 
  = 
  ; 
  and 
  this 
  being 
  so, 
  the 
  integration 
  may 
  be 
  

   regarded 
  as 
  extending 
  as 
  far 
  as 
  we 
  please 
  beyond 
  the 
  material 
  space 
  or 
  

   surface 
  (but 
  so 
  always 
  as 
  to 
  include 
  the 
  whole 
  of 
  the 
  material 
  space 
  or 
  

   surface) 
  — 
  for 
  instance, 
  in 
  the 
  case 
  of 
  a 
  spatial 
  distribution, 
  over 
  the 
  whole 
  

   (s 
  + 
  l)-dimensional 
  space; 
  and 
  in 
  the 
  case 
  of 
  a 
  superficial 
  distribution, 
  

   over 
  the 
  whole 
  of 
  the 
  s-dimeusional 
  surface 
  of 
  which 
  the 
  material 
  surface 
  

   is 
  a 
  part. 
  

  

  In 
  all 
  cases 
  of 
  surface-integrals 
  it 
  is, 
  unless 
  the 
  contrary 
  is 
  expressly 
  

  

  