﻿442 
  Mr. 
  J. 
  R. 
  Wilton 
  on 
  

  

  3. 
  In 
  the 
  particular 
  case 
  when 
  y 
  = 
  l'4, 
  its 
  approximate 
  

   value, 
  equation 
  (2) 
  becomes 
  

  

  and 
  in 
  this 
  form 
  it 
  may 
  be 
  solved 
  by 
  Darboux' 
  method, 
  and 
  

   — 
  though 
  with 
  so 
  much 
  labour 
  as 
  to 
  make 
  it 
  impracticable 
  — 
  

   the 
  solution 
  corresponding 
  to 
  any 
  initial 
  conditions 
  may 
  be 
  

   obtained. 
  Each 
  set 
  of 
  characteristics, 
  with 
  the 
  above 
  value 
  

   of 
  <y, 
  has 
  two 
  integrable 
  combinations, 
  namely, 
  

  

  d% 
  = 
  0, 
  dy 
  = 
  0, 
  

  

  dec 
  = 
  2ud 
  V 
  /(% 
  + 
  r 
  1 
  ), 
  dS 
  = 
  28 
  d 
  V 
  /(% 
  + 
  V 
  ) 
  ; 
  

   whence 
  

  

  « 
  = 
  (?+<?.wa 
  w 
  

  

  8 
  = 
  (Z+vWiv), 
  (5) 
  

  

  where 
  </> 
  and 
  -\|r 
  are 
  arbitrary 
  functions 
  of 
  their 
  arguments, 
  

   and 
  «, 
  ft, 
  7, 
  8 
  are 
  the 
  partial 
  derivatives 
  of 
  the 
  third 
  order. 
  

   (£> 
  v 
  , 
  a/t 
  v 
  denote 
  the 
  fifth 
  differential 
  coefficients 
  of 
  their 
  

   arguments. 
  The 
  general 
  solution 
  of 
  equation 
  (3) 
  is 
  therefore 
  

  

  v 
  = 
  (f+^)y'(f)-6(f 
  + 
  i7)^(f) 
  + 
  120(f) 
  

  

  + 
  (Z+ 
  yYV 
  (v)-G(Z+v)V(v) 
  + 
  I2f(v)- 
  . 
  (6) 
  

  

  We 
  have 
  to 
  choose 
  the 
  arbitrary 
  functions 
  <f> 
  and 
  i/r 
  so 
  as 
  

   to 
  satisfy 
  given 
  initial 
  conditions. 
  These 
  conditions 
  may 
  be 
  

   satisfied 
  by 
  making 
  the 
  surface 
  (6) 
  pass 
  through 
  a 
  given 
  

   curve, 
  and 
  touch 
  a 
  given 
  developable 
  all 
  along 
  that 
  curve. 
  

  

  We 
  may 
  take, 
  that 
  is 
  to 
  say, 
  f, 
  rj, 
  v, 
  —., 
  ^— 
  , 
  to 
  be, 
  on 
  this 
  

  

  curve, 
  given 
  functions 
  of 
  a 
  single 
  variable 
  A, 
  subject 
  to 
  the 
  

   condition 
  

  

  dn 
  "ftv 
  d£ 
  ^v 
  drj 
  

  

  (K 
  = 
  ~d%d\ 
  + 
  l$Y)dX 
  

   say 
  

  

  f 
  = 
  f 
  (\), 
  n 
  = 
  rj 
  {X), 
  v 
  = 
  v 
  {\), 
  gj=p<A). 
  ^ 
  v 
  = 
  9o[^), 
  

  

  where 
  , 
  c. 
  , 
  , 
  , 
  

  

  Vq 
  = 
  Po?o 
  + 
  9oVo 
  • 
  

  

  These 
  equations 
  will 
  give 
  us 
  the 
  values 
  of 
  r, 
  s, 
  t, 
  cc, 
  ft, 
  7, 
  f, 
  

  

  