﻿1875.] 
  

  

  Exact 
  Rectilinear 
  Motion 
  by 
  Linlcwork. 
  

  

  567 
  

  

  The 
  property 
  alluded 
  to 
  is 
  this 
  : 
  — 
  

  

  The 
  cosines 
  of 
  the 
  opposite 
  angles 
  of 
  any 
  quadrilateral 
  whose 
  sides 
  are 
  

   of 
  constant 
  length, 
  but 
  whose 
  angles 
  are 
  variable, 
  bear 
  a 
  linear 
  relation 
  to 
  

   each 
  other. 
  

  

  § 
  1. 
  In 
  fig. 
  1, 
  A 
  B 
  C 
  D 
  is 
  any 
  quadrilateral 
  of 
  which 
  the 
  sides 
  A 
  B, 
  

   BC, 
  CD, 
  DA 
  are 
  of 
  the 
  lengths 
  a, 
  b, 
  c, 
  d 
  respectively. 
  

  

  Kg. 
  1. 
  

  

  Then 
  it 
  is 
  clear 
  that 
  

  

  a 
  2 
  + 
  b 
  2 
  - 
  2ab 
  cos 
  B 
  = 
  c 
  2 
  + 
  d 
  2 
  - 
  2cd 
  cos 
  D 
  (1) 
  

  

  That 
  is, 
  there 
  is 
  a 
  linear 
  relation 
  of 
  the 
  most 
  general 
  character 
  between 
  

   the 
  cosines 
  of 
  the 
  variable 
  angles 
  B 
  and 
  D. 
  

  

  Before, 
  however, 
  this 
  property 
  can 
  be 
  taken 
  advantage 
  of 
  something 
  

   more 
  is 
  required; 
  the 
  angles 
  whose 
  cosines 
  bear 
  a 
  linear 
  relation 
  to 
  

   each 
  other 
  are 
  the 
  opposite 
  angles 
  of 
  a 
  closed 
  quadrilateral 
  ; 
  and 
  for 
  our 
  

   purpose 
  it 
  is 
  necessary 
  that 
  they 
  should 
  be 
  the 
  angles 
  at 
  the 
  base 
  of 
  an 
  

   open 
  trilateral 
  — 
  i. 
  e.,to 
  employ 
  the 
  language 
  of 
  linkwork, 
  the 
  angles 
  made 
  

   with 
  a 
  third 
  bar 
  by 
  two 
  bars 
  which 
  are 
  jointed 
  to 
  it. 
  To 
  effect 
  this 
  

   transformation 
  let 
  the 
  second 
  quadrilateral 
  A 
  |S 
  y 
  8 
  be 
  constructed 
  equal 
  

   in 
  every 
  respect 
  to 
  ABCD, 
  and 
  having 
  its 
  sides 
  £ 
  A, 
  /3 
  A 
  collinear 
  with 
  

   the 
  sides 
  B 
  A, 
  DAof 
  ABCD, 
  but 
  placed 
  in 
  a 
  reverse 
  position 
  so 
  as 
  to 
  

   be 
  the 
  image 
  of 
  ABCD. 
  This 
  new 
  quadrilateral 
  may 
  be 
  termed 
  the 
  

   "conjugate 
  image" 
  of 
  ABCD, 
  the 
  whole 
  figure 
  forming 
  what 
  may 
  be 
  

   termed 
  a 
  " 
  self 
  -con 
  jugate 
  sextilateral." 
  

  

  VOL. 
  XXIII. 
  2 
  u* 
  

  

  