﻿580 
  Mr. 
  F. 
  L. 
  Hitchcock 
  on 
  

  

  we 
  have 
  thus 
  the 
  three 
  parts 
  o£ 
  \/(jjt). 
  Combining 
  them 
  

   gives 
  

  

  V(<7T) 
  =tSV<T+ 
  (<£-<J>')t 
  + 
  S 
  . 
  tV<T 
  

  

  -ctSVt-(6> 
  + 
  6' 
  / 
  )o--S.c7Vt; 
  

   but 
  we 
  have, 
  identically, 
  

  

  V.W(T.T=(^-f)T, 
  .... 
  (8) 
  

  

  by 
  Tait, 
  § 
  186; 
  accordingly 
  the 
  first 
  three 
  terms 
  of 
  V(o"tJ 
  re- 
  

   duce 
  to 
  X/cr 
  . 
  t, 
  and 
  the 
  last 
  three, 
  similarly, 
  to 
  — 
  a 
  . 
  \7r 
  — 
  26a; 
  

   whence, 
  finally, 
  

  

  V(<7T)=Vo-.T-(7VT-20e7 
  (9) 
  

  

  It 
  may 
  be 
  noticed 
  that 
  — 
  6a 
  is 
  the 
  same 
  as 
  ScrV 
  • 
  t. 
  

  

  If 
  r 
  and 
  q 
  are 
  any 
  two 
  quaternion 
  functions 
  of 
  p 
  we 
  have 
  

  

  V(^)=Vtf. 
  r- 
  2 
  Vr 
  + 
  2S0 
  . 
  Vf 
  + 
  2S(V?V>, 
  • 
  (10) 
  

  

  which 
  follows 
  on 
  combining 
  (9) 
  with 
  V(Pcr), 
  &c, 
  and 
  which 
  

   the 
  reader 
  may 
  verify 
  with 
  ease. 
  

  

  5. 
  It 
  is 
  convenient 
  to 
  classify 
  vectors 
  by 
  the 
  effect 
  of 
  V 
  

   upon 
  them 
  : 
  if 
  Y\/<r 
  vanishes, 
  or 
  is 
  derivable 
  from 
  a 
  scalar 
  

   potential 
  and 
  its 
  distribution 
  is 
  irrotational 
  ; 
  if 
  S 
  \Jo~ 
  vanishes, 
  

   cr 
  is 
  derivable 
  from 
  a 
  vector 
  potential, 
  and 
  its 
  distribution 
  is 
  

   solenoidal; 
  while 
  if 
  both 
  these 
  conditions 
  are 
  fulfilled 
  at 
  once, 
  

   so 
  that 
  \Ja 
  = 
  0, 
  then 
  the 
  distribution 
  is 
  Laplacean. 
  These 
  

   distinctions 
  are 
  of 
  fundamental 
  importance 
  in 
  Physics. 
  

  

  There 
  are 
  also 
  vectors 
  which, 
  though 
  they 
  do 
  not 
  directly 
  

   satisfy 
  the 
  equation 
  VV°" 
  = 
  0, 
  yet 
  do 
  so 
  when 
  multiplied 
  by 
  

   a 
  variable 
  scalar. 
  Hamilton 
  and 
  Tait 
  showed 
  that 
  we 
  then 
  

   have 
  ScrVcr 
  = 
  0. 
  The 
  simplest 
  example 
  is 
  a 
  unit-vector 
  

   normal 
  to 
  a 
  series 
  of 
  surfaces, 
  and 
  capable, 
  therefore, 
  of 
  being 
  

   written 
  UVP. 
  

  

  Taking 
  the 
  two 
  vectors 
  VP 
  and 
  UVP, 
  we 
  shall 
  adopt 
  the 
  

   notation 
  : 
  

  

  dTJ\/F 
  = 
  dv=x 
  d 
  P> 
  

   d\/P 
  =d(tv)=yjrdp; 
  

  

  the 
  operators 
  y 
  and 
  % 
  are 
  then 
  vector 
  differentials, 
  functions 
  

   of 
  p, 
  and 
  always 
  linear 
  in 
  dp 
  ; 
  their 
  properties 
  appear 
  to 
  be 
  of 
  

   considerable 
  interest. 
  

  

  If 
  a. 
  and 
  ft 
  are 
  any 
  two 
  constant 
  unit-vectors, 
  we 
  shall 
  have 
  

  

  d 
  (£) 
  =_rfSaVP 
  ' 
  by(2) 
  ' 
  

  

  = 
  — 
  Sarf\7P, 
  because 
  a 
  is 
  constant, 
  

   = 
  — 
  Sayjrdp 
  ; 
  

  

  