﻿510 
  On 
  Non-linear 
  Partial 
  Differential 
  Equations. 
  [June 
  17, 
  

  

  are 
  obtained, 
  I 
  have 
  developed 
  more 
  in 
  detail 
  some 
  special 
  cases 
  of 
  

   interest. 
  

  

  For 
  the 
  convenience 
  of 
  the 
  reader, 
  I 
  have 
  in 
  § 
  1 
  briefly 
  recapitulated 
  

   the 
  principal 
  parts 
  of 
  the 
  two 
  papers 
  above 
  quoted. 
  In 
  § 
  2 
  I 
  have 
  given, 
  

   at 
  all 
  events, 
  a 
  first 
  sketch 
  of 
  a 
  general 
  theory 
  of 
  multiple 
  contact 
  with 
  

   quadrics 
  ; 
  in 
  § 
  3 
  the 
  particular 
  cases 
  of 
  three-, 
  four-, 
  five-, 
  and 
  six- 
  

   pointic 
  contact 
  are 
  discussed 
  ; 
  and 
  in 
  § 
  4 
  some 
  conditions 
  for 
  the 
  exist- 
  

   ence 
  of 
  points 
  of 
  four-, 
  five-, 
  six-pointic 
  single 
  (i. 
  e. 
  not 
  multiple) 
  contact 
  

   are 
  established. 
  

  

  Thus 
  far 
  the 
  investigation 
  concerns 
  the 
  contact 
  of 
  quadrics 
  only 
  with 
  

   other 
  surfaces. 
  The 
  concluding 
  part 
  of 
  the 
  paper 
  is 
  concerned 
  with 
  the 
  

   corresponding 
  problem 
  for 
  cubics, 
  in 
  which 
  case 
  conditions 
  of 
  possibility 
  

   do 
  not 
  arise 
  for 
  simple 
  or 
  two-pointic 
  contact, 
  but 
  are 
  first 
  met 
  with 
  for 
  

   three-pointic 
  contact. 
  The 
  conditions 
  in 
  question, 
  with 
  some 
  of 
  their 
  

   consequences, 
  are 
  here 
  given 
  ; 
  and 
  their 
  complexity 
  will 
  perhaps 
  be 
  

   sufficient 
  justification 
  for 
  not 
  pursuing 
  the 
  subject 
  further 
  in 
  this 
  

   direction. 
  

  

  VII. 
  " 
  On 
  the 
  Theory 
  of 
  the 
  Solution 
  of 
  a 
  System 
  of 
  Simultaneous 
  

   Non-linear 
  Partial 
  Differential 
  Equations 
  of 
  the 
  First 
  Order/'' 
  

   By 
  E. 
  J. 
  Nanson. 
  Communicated 
  by 
  Prof. 
  Cayley, 
  F.R.S. 
  

   Received 
  June 
  5_, 
  1875. 
  

  

  (Abstract.) 
  

  

  Given 
  an 
  equation 
  of 
  the 
  form 
  

  

  z=(f> 
  O 
  l5 
  a? 
  2 
  , 
  .... 
  ar 
  +m 
  , 
  a 
  v 
  a 
  2 
  , 
  . 
  . 
  . 
  a 
  J, 
  

   we 
  obtain 
  by 
  differentiation 
  with 
  respect 
  to 
  each 
  of 
  the 
  n+m 
  independent 
  

   variables 
  oc 
  v 
  ce 
  2 
  , 
  . 
  . 
  . 
  . 
  oc 
  n+m 
  , 
  and 
  elimination 
  of 
  the 
  n 
  arbitrary 
  constants 
  

   a 
  v 
  a 
  2 
  , 
  . 
  . 
  . 
  a 
  n 
  , 
  a 
  system 
  of 
  m 
  + 
  1 
  non-linear 
  partial 
  differential 
  equations 
  of 
  

   the 
  first 
  order. 
  Of 
  this 
  s} 
  r 
  stem 
  the 
  given 
  equation 
  may 
  be 
  said 
  to 
  be 
  a 
  

   " 
  complete 
  primitive." 
  

  

  Conversely, 
  given 
  a 
  system 
  of 
  non-linear 
  partial 
  differential 
  equations 
  

   of 
  the 
  first 
  order, 
  it 
  is 
  proposed 
  to 
  determine 
  the 
  conditions 
  which 
  must 
  

   be 
  satisfied 
  in 
  order 
  that 
  the 
  system 
  may 
  admit 
  of 
  a 
  complete 
  primitive, 
  

   and 
  also 
  to 
  examine 
  what 
  kind 
  of 
  solution, 
  if 
  any, 
  exists 
  when 
  the 
  con- 
  

   ditions 
  above 
  referred 
  to 
  are 
  not 
  satisfied. 
  

  

  The 
  late 
  Professor 
  Eoole 
  has 
  given 
  an 
  elegant 
  method 
  of 
  treating 
  a 
  

   system 
  of 
  linear 
  partial 
  differential 
  equations 
  of 
  the 
  first 
  order 
  ; 
  but 
  the 
  

   present 
  memoir 
  relates 
  to 
  a 
  more 
  general 
  system, 
  which 
  appears 
  not 
  to 
  

   have 
  been 
  hitherto 
  considered, 
  viz. 
  to 
  a 
  non-linear 
  system 
  of 
  partial 
  dif- 
  

   ferential 
  equations. 
  This 
  is 
  here 
  discussed 
  in 
  the 
  two 
  cases 
  — 
  first, 
  when 
  

   the 
  dependent 
  variable 
  z 
  is 
  not 
  explicitly 
  involved 
  in 
  the 
  proposed 
  

   system 
  ; 
  and, 
  secondly, 
  when 
  z 
  is 
  explicitly 
  involved 
  in 
  the 
  system, 
  the 
  

   solution 
  of 
  this 
  last 
  case 
  being 
  made 
  to 
  depend 
  upon 
  that 
  of 
  the 
  first- 
  

   mentioned 
  one. 
  

  

  