﻿584 
  Mr. 
  F. 
  L. 
  Hitchcock 
  on 
  

  

  special 
  case 
  of 
  (8), 
  

  

  (%-X')VVv=0, 
  

  

  which, 
  means 
  that 
  % 
  and 
  its 
  conjugate 
  have 
  the 
  same 
  effect 
  

   on 
  y\Jv. 
  But 
  it 
  was 
  shown 
  in 
  Art. 
  6 
  that 
  % 
  turns 
  every 
  

   vector 
  into 
  a 
  certain 
  plane, 
  and 
  %' 
  turns 
  every 
  vector 
  into 
  

   another 
  plane 
  ; 
  hence 
  x^V 
  v 
  ^ 
  es 
  a 
  l° 
  n 
  g 
  the 
  line 
  of 
  intersection 
  

   of 
  these 
  two 
  planes. 
  

  

  If 
  v' 
  be 
  a 
  unit- 
  vector 
  such 
  that 
  yy 
  l 
  ' 
  = 
  0, 
  it 
  follows 
  that 
  

  

  <XySJv=oeVvv' 
  '; 
  

  

  to 
  determine 
  the 
  unknown 
  scalar 
  #, 
  take 
  e 
  and 
  ?7 
  two 
  unit- 
  

   vectors 
  such 
  that 
  %e 
  = 
  <76 
  and 
  yri=g 
  x 
  i)\ 
  it 
  may 
  he 
  easily 
  

   shown 
  that 
  e, 
  ?;, 
  and 
  v 
  will 
  then 
  form 
  a 
  rectangular 
  system 
  

   (see 
  Ex. 
  2); 
  and 
  they 
  may 
  be 
  taken 
  so 
  that 
  erj 
  = 
  v. 
  It 
  is 
  

  

  then 
  legitimate 
  to 
  write 
  —- 
  in 
  the 
  following 
  form 
  (Tait, 
  

   § 
  176): 
  dn 
  

  

  SiV. 
  %v= 
  —ge$ev 
  r 
  — 
  g\r)$r]v'\ 
  

   operate 
  by 
  Vv, 
  

  

  i 
  SjWVv= 
  — 
  grjSev 
  1 
  +gi€$r)v', 
  

  

  and 
  by 
  using 
  again 
  the 
  same 
  form 
  of 
  %, 
  

  

  where 
  m 
  l 
  is 
  the 
  coefficient 
  of 
  % 
  in 
  the 
  strain-cubic. 
  Thus 
  if 
  

   I 
  be 
  the 
  angle 
  between 
  v 
  and 
  v', 
  the 
  tensor 
  of 
  ^VV^ 
  is 
  

   m 
  1 
  tan 
  £. 
  

  

  9. 
  If, 
  further, 
  a 
  be 
  any 
  vector 
  in 
  the 
  tangent 
  plane, 
  so 
  

   that 
  at 
  all 
  points 
  S<7j/ 
  = 
  0, 
  then 
  by 
  (8) 
  

  

  (0-(/>> 
  + 
  VvVW=O; 
  

   here 
  (f>v 
  may 
  be 
  written 
  y- 
  ; 
  by 
  (5) 
  we 
  obtain 
  

  

  VS<7V 
  = 
  0=-%'er- 
  fiv; 
  

   the 
  values 
  of 
  $v 
  and 
  <£V 
  give 
  by 
  substituting, 
  — 
  

  

  i+y+VvVV=0, 
  (20) 
  

  

  provided 
  the 
  operand 
  be 
  at 
  right 
  angles 
  to 
  v. 
  

  

  