﻿568 
  

  

  Mr. 
  A. 
  B. 
  Kempe 
  on 
  the 
  production 
  of 
  [June 
  17, 
  

  

  It 
  is 
  clear 
  that 
  the 
  angle 
  $ 
  is 
  equal 
  to 
  the 
  angle 
  D 
  ; 
  thus 
  we 
  have 
  the 
  

   sides 
  B 
  C, 
  of 
  the 
  open 
  trilateral 
  C 
  B 
  S 
  y 
  making 
  angles 
  with 
  A 
  B 
  whose 
  

   cosines 
  bear 
  a 
  linear 
  relation 
  to 
  each 
  other 
  however 
  the 
  figure 
  be 
  de- 
  

   formed. 
  

  

  Since, 
  however, 
  the 
  relation 
  is 
  an 
  angle 
  relation, 
  it 
  is 
  unnecessary 
  that 
  

   the 
  conjugate 
  image 
  should 
  be 
  equal 
  to 
  the 
  original 
  quadrilateral 
  ; 
  for 
  if 
  

   the 
  figure 
  A 
  B' 
  C 
  D' 
  be 
  constructed 
  similar 
  to 
  A 
  /3 
  y 
  8 
  the 
  angle 
  D' 
  is 
  

   clearly 
  equal 
  to 
  the 
  angle 
  8, 
  and 
  we 
  have 
  the 
  sides 
  C 
  B, 
  C 
  D' 
  making 
  

   angles 
  with 
  A 
  B 
  whose 
  cosines 
  bear 
  a 
  linear 
  relation 
  to 
  each 
  other. 
  This 
  

   makes 
  our 
  results 
  more 
  general 
  ; 
  and 
  we 
  are 
  moreover 
  able 
  to 
  make 
  the 
  

   points 
  D 
  and 
  B', 
  or 
  the 
  points 
  D' 
  and 
  B, 
  coincide 
  if 
  necessary. 
  This 
  

   more 
  general 
  form 
  of 
  figure, 
  consisting 
  of 
  two 
  quadrilaterals, 
  one 
  of 
  which 
  

   is 
  the 
  enlarged 
  or 
  reduced 
  positive 
  or 
  negative 
  image 
  of 
  the 
  other, 
  may 
  

   still 
  be 
  appropriately 
  termed 
  a 
  " 
  self- 
  conjugate 
  sextilateral," 
  the 
  qua- 
  

   drilaterals 
  being 
  still 
  called 
  the 
  one 
  the 
  " 
  self 
  -conjugate 
  image" 
  of 
  the 
  

   other. 
  

  

  § 
  2. 
  Now 
  let 
  the 
  linkage 
  in 
  fig. 
  2 
  be 
  constructed, 
  

  

  rig. 
  2. 
  

  

  

  

  

  

  

  

  L 
  

  

  \ 
  <•'<'. 
  

  

  

  B 
  

  

  IN" 
  . 
  

  

  A' 
  

  

  in 
  which 
  AB=«, 
  AB'=k«, 
  

  

  BC 
  = 
  b, 
  B'C=kb, 
  

  

  CD=c, 
  CT>'=kc, 
  

  

  DA=rf, 
  B'A=Jcd, 
  

  

  Jc 
  being 
  positive 
  or 
  negative, 
  and 
  greater, 
  equal 
  to, 
  or 
  less 
  than 
  unity, 
  so 
  

   that 
  the 
  linkage 
  forms 
  a 
  self 
  -conjugate 
  sextilateral, 
  the 
  quadrilaterals 
  

   A 
  B 
  C 
  D, 
  A 
  B' 
  D' 
  being 
  self 
  -conjugate 
  images 
  the 
  one 
  of 
  the 
  other. 
  

   Now 
  take 
  any 
  point 
  P 
  on 
  B 
  C, 
  and 
  let 
  BP=A, 
  and 
  take 
  a 
  point 
  P' 
  on 
  

  

  cd 
  

  

  D' 
  C 
  such 
  that 
  D' 
  P'=A^ 
  . 
  Draw 
  P 
  N 
  and 
  P' 
  N' 
  perpendicular 
  to 
  A 
  B. 
  

   Then 
  

  

  B 
  N 
  = 
  X 
  cos 
  B, 
  

  

  D' 
  N=\ 
  c 
  4 
  cos 
  D' 
  = 
  cos 
  D, 
  

  

  