﻿Vector 
  Differentials. 
  579 
  

  

  We 
  have 
  thus 
  the 
  means 
  of 
  finding 
  the 
  effect 
  of 
  V 
  on 
  any 
  

   function, 
  scalar 
  or 
  vector, 
  by 
  merely 
  differentiating 
  it. 
  

  

  4. 
  The 
  following 
  useful 
  formulas 
  will 
  be 
  familiar 
  to 
  students 
  

   ofTait:— 
  

  

  V(FP)=^VP, 
  

   and 
  

  

  V(PP 
  1 
  )=P 
  1 
  VP 
  + 
  PVP 
  1 
  , 
  

  

  in 
  which 
  the 
  order 
  is 
  not 
  important, 
  and 
  also 
  

  

  V(P<x) 
  = 
  VP.<r 
  + 
  PVcr, 
  

  

  where 
  the 
  order 
  is 
  vital. 
  Here 
  P 
  and 
  P 
  x 
  are 
  scalars, 
  FP 
  is 
  a 
  

   scalar 
  function 
  of 
  P, 
  and 
  o 
  is 
  a 
  vector. 
  

  

  To 
  find 
  the 
  effect 
  of 
  y 
  on 
  the 
  product 
  of 
  any 
  two 
  vectors 
  

   <t 
  and 
  t 
  we 
  may 
  adopt 
  the 
  notation 
  do 
  = 
  (j>dp 
  and 
  dr 
  = 
  ddp 
  ; 
  

   whence 
  

  

  d(oT) 
  = 
  do 
  .t 
  + 
  o 
  .dr 
  

  

  = 
  falp 
  . 
  t 
  + 
  cr 
  . 
  6dp, 
  

  

  From 
  the 
  scalar 
  part 
  of 
  this 
  differential 
  we 
  have 
  

  

  d&OT=&dp{6 
  f 
  T+6'o), 
  

  

  whence 
  by 
  (1), 
  

  

  VS<rr=-<*/T-0'«r; 
  (5) 
  

  

  and 
  from 
  the 
  vector 
  part, 
  

  

  d 
  Not 
  = 
  Yoddp 
  - 
  Vr<l>dp, 
  

  

  each 
  term 
  of 
  which, 
  by 
  the 
  last 
  article, 
  contributes 
  its 
  portion 
  

   of 
  VVctt. 
  If 
  we 
  take 
  <£ 
  t 
  = 
  Vcr#, 
  we 
  have 
  

  

  <h 
  = 
  Nodi 
  +jV<rdj 
  4- 
  kY<rdJc 
  

  

  = 
  _S 
  . 
  o-Vr- 
  o-SVt-00-, 
  (Tait, 
  §§ 
  89, 
  90) 
  

  

  by 
  the 
  ordinary 
  transformations. 
  

  

  Similarly 
  the 
  part 
  corresponding 
  to 
  — 
  Yrcjidp 
  is 
  

  

  -f 
  S 
  . 
  tW 
  + 
  tS 
  V<7 
  4- 
  c/)T 
  : 
  

   by 
  adding 
  the 
  vector 
  parts 
  of 
  these 
  two 
  quaternions 
  we 
  have 
  

   VVV<7t=tSV0--0Wt 
  + 
  (/>t-0<t, 
  ... 
  (6) 
  

   and 
  by 
  adding 
  the 
  scalar 
  parts, 
  

  

  SV(<7t)=S.tV<7-S.<7Vt; 
  .... 
  (7) 
  

  

  