﻿Vector 
  Differentials. 
  585 
  

  

  Operating 
  on 
  Y\/v 
  and 
  substituting 
  the 
  result 
  in 
  (18 
  a) 
  

   gives 
  

  

  V.vVSVv=(^-x)vVv. 
  . 
  . 
  . 
  (18 
  6) 
  

   Again, 
  by 
  using 
  the 
  value 
  of 
  Y\/v 
  from 
  (15), 
  

  

  ^ 
  TTV7 
  V 
  $ 
  V 
  

  

  rtn 
  an* 
  

  

  and 
  this, 
  combined 
  with 
  the 
  result 
  o£ 
  the 
  last 
  article, 
  gives 
  

  

  V.vVSVv=V^^-rVS-Vv). 
  . 
  . 
  (18c) 
  

  

  One 
  other 
  transformation 
  is 
  obtained 
  from 
  the 
  yy 
  of 
  the 
  

   last 
  article 
  by 
  putting 
  cov'— 
  — 
  givSyv'— 
  #eSey', 
  so 
  that 
  the 
  com- 
  

   ponent 
  of 
  v' 
  at 
  right 
  angles 
  to 
  v 
  is 
  SiA> 
  . 
  w^yv, 
  and 
  this 
  gives, 
  

   by 
  substituting 
  in 
  (18 
  c), 
  — 
  

  

  ▼<£-^K-VB7r)-p 
  s 
  . 
  . 
  (18,0 
  

  

  that 
  is, 
  the 
  vector 
  in 
  parentheses 
  is 
  normal 
  to 
  the 
  surface 
  

   P 
  = 
  const. 
  Here 
  it 
  is 
  noteworthy 
  that 
  both 
  the 
  vector 
  

   Yv\/$\Jv 
  and 
  the 
  linear 
  and 
  vector 
  function 
  m^ 
  - 
  1 
  are 
  nume- 
  

   rically 
  determinate 
  all 
  over 
  a 
  given 
  surface 
  P 
  = 
  P 
  . 
  Thus 
  

   (18) 
  shows 
  the 
  character 
  of 
  v, 
  provided 
  y 
  2 
  P 
  = 
  0, 
  in 
  the 
  

   immediate 
  neighbourhood 
  of 
  the 
  given 
  surface. 
  

  

  If 
  v 
  be 
  so 
  given 
  as 
  to 
  satisfy 
  (18), 
  P 
  may 
  be 
  written 
  V 
  _1 
  (^), 
  

   and 
  is 
  determined 
  by 
  (19), 
  since 
  u=-\ogt 
  by 
  (17). 
  

  

  Examples. 
  

  

  1. 
  Give 
  in 
  terms 
  of 
  y 
  the 
  curvature 
  of 
  a 
  normal 
  section 
  of 
  

   the 
  surface 
  P 
  = 
  const. 
  (Tait, 
  § 
  350, 
  where 
  v 
  is 
  the 
  tv 
  of 
  this 
  

   paper.) 
  

  

  2. 
  Show 
  that 
  two 
  of 
  the 
  roots 
  of 
  the 
  cubic 
  in 
  y 
  correspond 
  

   to 
  the 
  sections 
  of 
  greatest 
  and 
  least 
  curvature. 
  

  

  3. 
  If 
  v 
  1 
  correspond 
  to 
  the 
  other 
  root, 
  show 
  that 
  if 
  v, 
  v' 
  and 
  

   yv 
  are 
  coplanar, 
  yfv 
  is 
  parallel 
  to 
  yv. 
  Of 
  what 
  class 
  of 
  

   surfaces 
  is 
  this 
  a 
  property 
  ? 
  

  

  4. 
  Show 
  that 
  if 
  P 
  is 
  a 
  homogeneous 
  function 
  of 
  x, 
  y, 
  

   and 
  z, 
  any 
  straight 
  line 
  through 
  the 
  origin 
  cuts 
  the 
  surfaces 
  

   denoted 
  by 
  P 
  at 
  a 
  constant 
  angle. 
  

  

  5. 
  Show 
  that 
  if 
  P 
  is 
  a 
  homogeneous 
  spherical 
  harmonic, 
  

   V~UVv 
  • 
  v) 
  = 
  const, 
  is 
  the 
  equation 
  of 
  a 
  cone. 
  

  

  