﻿Nodal- 
  Slide 
  Method 
  of 
  Focometry. 
  23 
  

  

  points, 
  Ni 
  and 
  N 
  2 
  the 
  nodal 
  points, 
  F 
  T 
  and 
  F 
  2 
  the 
  principal 
  

   foci. 
  Suppose 
  the 
  system 
  to 
  be 
  rotated 
  about 
  through 
  a 
  

   small 
  angle 
  6 
  so 
  that 
  the 
  principal 
  axis 
  moves 
  into 
  the 
  

   position 
  indicated 
  by 
  the 
  dotted 
  line. 
  Then, 
  to 
  a 
  first 
  approxi- 
  

   mation, 
  the 
  nodal 
  points 
  Nj 
  and 
  N 
  2 
  will 
  move 
  into 
  the 
  positions 
  

   Ni' 
  and 
  N 
  2 
  ' 
  and 
  the 
  image 
  P 
  2 
  Q 
  2 
  will 
  move 
  in 
  the 
  same 
  plane 
  

   into 
  the 
  position 
  P 
  2 
  'Q 
  2 
  ' 
  obtained 
  by 
  drawing 
  N 
  2 
  'P 
  2 
  ' 
  and 
  N 
  2 
  'Q 
  2/ 
  

   parallel 
  to 
  P^NY 
  nnd 
  Qil^Y 
  respectively. 
  

  

  Let 
  ^ 
  x 
  O 
  = 
  l, 
  N 
  2 
  N 
  1= 
  a, 
  NjP^m, 
  N 
  2 
  P 
  2 
  =t>. 
  

  

  Then 
  ISJS^W 
  and 
  N 
  2 
  N 
  2 
  ' 
  = 
  {a 
  + 
  1)6. 
  

  

  Hence 
  the 
  displacement 
  o£ 
  the 
  image 
  is 
  given 
  by 
  

  

  S 
  =Q 
  2 
  Q 
  2 
  / 
  =N 
  2 
  N 
  2 
  '+M?.N 
  1 
  N/ 
  

  

  = 
  (a+l)0--.lO 
  

  

  u 
  

  

  = 
  {a 
  + 
  l(l-m)}0, 
  (1) 
  

  

  where 
  ??i=- 
  , 
  the 
  magnification. 
  

   u 
  

  

  In 
  order 
  that 
  the 
  displacement 
  of 
  the 
  image 
  may 
  be 
  zero 
  

  

  for 
  a 
  oiven 
  value 
  of 
  6 
  we 
  must 
  have 
  

  

  s=a 
  + 
  l(l— 
  ni)=0 
  3 
  

  

  or 
  a 
  + 
  1 
  , 
  . 
  

  

  ™^— 
  j- 
  (2) 
  

  

  There 
  is 
  thus 
  one, 
  and 
  only 
  one, 
  position 
  of 
  the 
  axis 
  of 
  rotation 
  

  

  for 
  which 
  there 
  will 
  be 
  no 
  displacement 
  of 
  the 
  image, 
  viz. 
  

  

  that 
  which 
  divides 
  the 
  distance 
  between 
  the 
  nodal 
  points 
  

  

  externally 
  in 
  a 
  ratio 
  equal 
  to 
  the 
  value 
  of 
  the 
  magnification 
  — 
  

  

  a 
  result 
  obtained 
  in 
  another 
  way 
  by 
  Prof. 
  Anderson. 
  The 
  

  

  best 
  position 
  for 
  the 
  axis 
  of 
  rotation 
  will, 
  as 
  already 
  pointed 
  

  

  ds 
  

   out, 
  be 
  that 
  for 
  which 
  -— 
  . 
  is 
  a 
  maximum 
  subject 
  to 
  the 
  con- 
  

  

  Oil) 
  

  

  dition 
  given 
  by 
  equation 
  (2). 
  

  

  Differentiating 
  (1) 
  and 
  substituting 
  from 
  (2), 
  we 
  get 
  

  

  *-d-*-4 
  ..... 
  (3) 
  

  

  The 
  rate 
  of 
  change 
  of 
  the 
  displacement 
  thus 
  varies 
  directly 
  

   as 
  a, 
  the 
  distance 
  between 
  the 
  nodal 
  points 
  and 
  inversely 
  as 
  /, 
  

   the 
  distance 
  of 
  the 
  nul 
  point 
  from 
  the 
  first 
  nodal 
  point. 
  

  

  It 
  is 
  greatest 
  when 
  2 
  = 
  0, 
  i. 
  e. 
  when 
  m 
  = 
  oo, 
  and 
  the 
  nul 
  

   point 
  coincides 
  with 
  N 
  x 
  . 
  It 
  thus 
  appears 
  that 
  the 
  best 
  

  

  