﻿582 
  Mr. 
  F. 
  L. 
  Hitchcock 
  on 
  

  

  VvV 
  are 
  mutually 
  at 
  right 
  angles, 
  while 
  T^v=TVV» 
  / 
  . 
  These 
  

   facts 
  are 
  expressed 
  by 
  the 
  equation 
  

  

  v 
  X 
  v 
  = 
  NSJv. 
  ...... 
  (15) 
  

  

  Writing, 
  as 
  above, 
  — 
  for 
  differentiation 
  along 
  the 
  normal 
  

  

  to 
  the 
  surface 
  P 
  = 
  const, 
  we 
  shall 
  have 
  

  

  = 
  — 
  Svtydp 
  — 
  S 
  (tv) 
  xdp 
  ; 
  

  

  the 
  last 
  term 
  vanishes 
  by 
  (13), 
  and 
  yjr 
  is 
  self-conjugate 
  by 
  

   (11), 
  hence 
  

  

  vgW-ivr, 
  (16) 
  

  

  an 
  equation 
  which 
  should 
  be 
  compared 
  with 
  (5) 
  and 
  with 
  

   (12), 
  from 
  the 
  former 
  of 
  which 
  it 
  may 
  be 
  deduced 
  by 
  

   applying 
  (14). 
  

  

  7. 
  We 
  are 
  now 
  able 
  to 
  examine 
  the 
  criterion 
  that 
  the 
  

   vector 
  v 
  shall, 
  besides 
  being 
  derivable 
  from 
  a 
  scalar 
  potential 
  

   by 
  means 
  of 
  a 
  scalar 
  factor, 
  be 
  derivable 
  from 
  one 
  particular 
  

   scalar 
  potential 
  which 
  shall 
  satisfy 
  Laplace's 
  equation 
  ; 
  to 
  

   find, 
  in 
  other 
  words, 
  the 
  condition 
  that 
  a 
  scalar 
  t 
  can 
  be 
  found 
  

  

  such 
  that 
  V(^) 
  = 
  V 
  2p 
  = 
  °- 
  

  

  dP 
  

   Remembering 
  that 
  -7- 
  =£, 
  we 
  shall 
  have 
  

  

  & 
  dn 
  ' 
  

  

  yfrdp 
  = 
  d(tv) 
  

  

  = 
  vdt 
  + 
  tdv 
  

   =—v8dpVt 
  + 
  txdp 
  

  

  = 
  -vBdp 
  d 
  -^ 
  + 
  t 
  X 
  dp, 
  by 
  (16), 
  

  

  = 
  ~ 
  dn~ 
  v 
  ® 
  vd 
  P- 
  tv 
  ®W 
  d 
  P 
  + 
  *X 
  d 
  P 
  

  

  = 
  — 
  -j- 
  v$vdp 
  — 
  txv$vdp 
  + 
  ty'dp, 
  

  

  where 
  the 
  last 
  step 
  follows 
  because 
  yfr 
  is 
  self-conjugate. 
  

  

  By 
  inspection 
  of 
  this 
  result 
  it 
  is 
  evident 
  that, 
  upon 
  any 
  

   vector 
  in 
  the 
  tangent 
  plane, 
  the 
  strain 
  yjr 
  has 
  the 
  same 
  effect 
  

   as 
  %', 
  with 
  the 
  sole 
  difference 
  that 
  -ty 
  allongates 
  the 
  vector 
  by 
  

   the 
  factor 
  t. 
  There 
  are 
  important 
  geometrical 
  applications 
  of 
  

   this 
  fact, 
  some 
  of 
  which 
  will 
  be 
  found 
  in 
  the 
  examples 
  at 
  the 
  

  

  