﻿578 
  Mr. 
  F. 
  L. 
  Hitchcock 
  on 
  

  

  We. 
  have 
  also 
  

  

  dF=-SdpVP 
  

  

  = 
  + 
  Sdp(«StVP 
  +/S/VP 
  + 
  kSky?) 
  

   dV 
  .dP 
  , 
  dP> 
  

  

  c 
  . 
  / 
  .ar 
  .ar 
  , 
  ar\ 
  , 
  

  

  0y 
  

  

  or 
  , 
  , 
  dv 
  , 
  dp, 
  

  

  = 
  y— 
  ax 
  + 
  ^— 
  ay 
  + 
  - 
  1 
  - 
  dz, 
  

   ax 
  dy 
  ,J 
  dz 
  

  

  because 
  dx= 
  — 
  Siop, 
  &c. 
  

  

  From 
  (1) 
  it 
  appears 
  that 
  if 
  oT 
  be 
  given, 
  yP 
  can 
  be 
  written 
  

   by 
  inspection. 
  

  

  3. 
  Taking 
  next 
  ar 
  any 
  vector 
  function 
  of 
  p, 
  we 
  have 
  

  

  da 
  = 
  (f>dp, 
  (3) 
  

  

  where 
  <f> 
  is 
  a 
  linear 
  and 
  vector 
  function. 
  And, 
  directly, 
  

  

  do- 
  , 
  , 
  . 
  

  

  dh=* 
  e 
  w 
  

  

  e 
  

  

  For 
  a 
  fascinating 
  account 
  of 
  the 
  various 
  types 
  of 
  these 
  

   functions, 
  see 
  the 
  last 
  chapter 
  of 
  Kelland 
  and 
  Tait's 
  ' 
  Intro- 
  

   duction 
  to 
  Quaternions/ 
  The 
  function 
  is 
  there 
  considered 
  

   as 
  a 
  homogeneous 
  strain, 
  and 
  it 
  seems 
  convenient 
  so 
  to 
  speak 
  

   of 
  it, 
  even 
  in 
  those 
  cases 
  where 
  it 
  could 
  not 
  exist 
  in 
  a 
  physical 
  

   sense 
  ; 
  for 
  example, 
  when 
  the 
  sum 
  of 
  the 
  roots 
  of 
  the 
  strain- 
  

   cubic 
  is 
  zero. 
  

  

  To 
  show 
  that 
  X/a 
  may 
  be 
  written 
  by 
  inspection 
  when 
  da 
  is 
  

   given, 
  we 
  may 
  put 
  

  

  q 
  = 
  i<j)i 
  +j<j>j 
  + 
  k(f>k 
  ; 
  

  

  and, 
  if 
  <f> 
  consists 
  of 
  several 
  terms, 
  we 
  may 
  consider 
  each 
  ol 
  

   them 
  as 
  a 
  separate 
  linear 
  and 
  vector 
  function, 
  call 
  them 
  

   0i; 
  02; 
  & 
  c 
  - 
  9 
  to 
  these 
  will 
  correspond 
  q 
  u 
  q 
  2 
  , 
  . 
  . 
  ., 
  whose 
  sum, 
  

   since 
  q 
  is 
  linear 
  in 
  $, 
  must 
  give 
  the 
  q 
  of 
  the 
  whole 
  function 
  <p. 
  

  

  But 
  \/a=q, 
  by 
  (4) 
  ; 
  thus 
  we 
  can 
  write 
  down 
  X/cr 
  if 
  we 
  know 
  

   the 
  part 
  of 
  q 
  contributed 
  by 
  each 
  term_of 
  <£. 
  

  

  Taking 
  special 
  cases, 
  a 
  term 
  of 
  the 
  form 
  £Sa\, 
  which 
  we 
  

   may 
  call 
  <£ 
  l5 
  and 
  where 
  X 
  is 
  any 
  vector 
  whatever, 
  gives 
  

  

  g 
  l 
  = 
  ip&eti 
  -+-j/3Saj 
  + 
  k/3$ak 
  

  

  and 
  in 
  a 
  similar 
  manner 
  the 
  forms 
  V«X, 
  VaX/9, 
  and 
  g\ 
  give, 
  

   in 
  order, 
  2a, 
  Sa/5, 
  and 
  — 
  3#. 
  Any 
  other 
  terms 
  that 
  may 
  

   occur 
  are 
  to 
  be 
  treated 
  in 
  this 
  way, 
  and 
  the 
  sum 
  of 
  the 
  results 
  

   taken. 
  

  

  