﻿1875.] 
  

  

  Prof. 
  Cayley 
  on 
  Prepotentials. 
  

  

  417 
  

  

  William 
  Archer, 
  M.E.I. 
  A. 
  

   James 
  Bisdon 
  Bennett, 
  M.D. 
  

   Dietrich 
  Brandis, 
  Ph.D., 
  E.L.S. 
  

   James 
  Caird, 
  C.B. 
  

   Prof. 
  John 
  Casey, 
  LL.D. 
  

   August 
  Dupre, 
  Ph.D., 
  F.6.S. 
  

   James 
  Geikie, 
  E.B.S.E. 
  

   James 
  Whitbread 
  Lee 
  Glaisher, 
  

   M.A. 
  

  

  John 
  Baboneau 
  Nickterlien 
  Hen- 
  

   nessey, 
  F.B.A.S. 
  

  

  Emanuel 
  Klein, 
  M.D. 
  

  

  E. 
  Eay 
  Lankester, 
  M.A. 
  

  

  George 
  Strong 
  Nares, 
  Capt. 
  B.N. 
  

  

  Eobert 
  Stirling 
  JNTewall, 
  E.E.A.S. 
  

  

  William 
  Chandler 
  Eoberts, 
  E.C.S. 
  

  

  Major-General 
  Henry 
  Y. 
  D. 
  Scott, 
  

   E.E., 
  C.B. 
  

  

  Thanks 
  were 
  given 
  to 
  the 
  Scrutators. 
  

  

  June 
  10, 
  1875. 
  

  

  JOSEPH 
  DALTON 
  HOOKER, 
  C.B., 
  President, 
  in 
  the 
  Chair. 
  

  

  The 
  Presents 
  received 
  were 
  laid 
  on 
  the 
  table, 
  and 
  thanks 
  ordered 
  for 
  

   them. 
  

  

  Dr. 
  James 
  Eisdon 
  Beunett, 
  Mr. 
  James 
  Caird, 
  Mr. 
  James 
  Whitbread 
  

   Lee 
  Glaisher, 
  Mr. 
  J. 
  Baboneau 
  Nickterlien 
  Hennessey, 
  Mr. 
  Williarn. 
  

   Chandler 
  Eoberts, 
  and 
  Major-General 
  Henry 
  Youug 
  Darracote 
  Scott 
  

   were 
  admitted 
  into 
  the 
  Society. 
  

  

  The 
  following 
  Papers 
  were 
  read 
  : 
  — 
  

  

  I. 
  " 
  A 
  Memoir 
  on 
  Prepotentials." 
  By 
  Prof. 
  Cayley, 
  F.E.S. 
  

   Received 
  April 
  8, 
  1875. 
  

  

  (Abstract.) 
  

  

  The 
  present 
  memoir 
  relates 
  to 
  multiple 
  integrals 
  expressed 
  in 
  terms 
  of 
  

   the 
  + 
  ultimately 
  disappearing 
  variables 
  {x..z, 
  w), 
  and 
  the 
  same 
  

   number 
  of 
  parameters 
  (a 
  . 
  . 
  c, 
  e), 
  and 
  being 
  of 
  the 
  form 
  

  

  C 
  p 
  dnr 
  

  

  where 
  p 
  and 
  dvr 
  depend 
  only 
  on 
  the 
  variables 
  (a? 
  . 
  . 
  z, 
  w). 
  Such 
  an 
  inte- 
  

   gral, 
  in 
  regard 
  to 
  the 
  index 
  |s 
  + 
  g, 
  is 
  said 
  to 
  be 
  " 
  prepotential," 
  and 
  in 
  the 
  

   particular 
  case 
  q 
  = 
  — 
  | 
  to 
  be 
  "potential." 
  

  

  I 
  use 
  throughout 
  the 
  language 
  of 
  hyper-tridimensional 
  geometry 
  : 
  

   (x 
  . 
  . 
  z, 
  iv) 
  and 
  (a 
  . 
  . 
  c, 
  e) 
  are 
  regarded 
  as 
  coordinates 
  of 
  points 
  in 
  (s 
  + 
  1)- 
  

   dimensional 
  space, 
  the 
  former 
  of 
  them 
  determining 
  the 
  position 
  of 
  an 
  

   element 
  pd™ 
  of 
  attracting 
  matter, 
  the 
  latter 
  being 
  the 
  attracted 
  point 
  ; 
  

   viz. 
  we 
  have 
  a 
  mass 
  of 
  matter 
  =J'pd>m 
  distributed 
  in 
  such 
  manner 
  that 
  

   d>m 
  beiug 
  the 
  element 
  of 
  (s+ 
  1)- 
  or 
  lower-dimensional 
  volume 
  at 
  the 
  

  

  vol. 
  xxiii. 
  2 
  is 
  

  

  