﻿484 
  Prof. 
  J. 
  D. 
  Everett 
  on 
  Focal 
  Lines, 
  

  

  point 
  on 
  the 
  revolving 
  circle, 
  PQ 
  a 
  perpendicular 
  on 
  the 
  axis 
  

   of 
  revolution, 
  and 
  6 
  the 
  inclination 
  of 
  the 
  straight 
  line 
  PAS 
  

   to 
  PQ 
  or 
  AO. 
  

  

  The 
  two 
  focal 
  lines 
  are 
  always 
  at 
  the 
  centres 
  of 
  curvature 
  

   of 
  two 
  mutually 
  rectangular 
  normal 
  sections 
  of 
  the 
  wave- 
  

   front, 
  one 
  being 
  the 
  section 
  of 
  greatest 
  and 
  the 
  other 
  of 
  least 
  

   radius 
  of 
  curvature. 
  For 
  an 
  element 
  of 
  the 
  wave-front 
  at 
  P 
  

   one 
  of 
  these 
  sections 
  is 
  the 
  circle 
  shown 
  in 
  our 
  diagram, 
  and 
  

   PA 
  is 
  its 
  radius 
  of 
  curvature. 
  The 
  other 
  is 
  the 
  section 
  of 
  

   the 
  tore 
  made 
  by 
  a 
  plane 
  through 
  PAS 
  perpendicular 
  to 
  the 
  

   plane 
  of 
  the 
  diagram. 
  To 
  find 
  its 
  radius 
  of 
  curvature 
  note 
  

   that 
  QP 
  is 
  the 
  radius 
  of 
  a 
  circular 
  section 
  made 
  by 
  a 
  plane 
  

   perpendicular 
  to 
  the 
  axis 
  QS 
  ; 
  hence, 
  bv 
  Meunier's 
  theorem, 
  

   the 
  required 
  radius 
  of 
  curvature 
  is 
  PQ 
  sec 
  0, 
  that 
  is 
  PS. 
  

   The 
  two 
  focal 
  lines 
  are 
  accordingly 
  at 
  A 
  and 
  S. 
  A 
  is 
  called 
  

   the 
  primary, 
  and 
  S 
  the 
  secondary 
  focus. 
  If 
  we 
  make 
  P 
  travel 
  

   round 
  the 
  circle 
  shown 
  in 
  the 
  figure 
  the 
  primary 
  focus 
  re- 
  

   mains 
  fixed 
  at 
  A, 
  and 
  the 
  secondary 
  focus 
  travels 
  along 
  the 
  

   axis 
  QS, 
  its 
  distance 
  from 
  being 
  OA 
  sec 
  6, 
  which 
  runs 
  

   from 
  zero 
  to 
  infinity 
  in 
  both 
  directions. 
  On 
  the 
  other 
  hand, 
  

   if 
  we 
  make 
  P 
  revolve 
  round 
  the 
  axis 
  QS 
  the 
  secondary 
  focus 
  

   remains 
  fixed 
  at 
  S, 
  and 
  the 
  primary 
  focus 
  generates 
  the 
  

   circular 
  axis 
  of 
  the 
  tore. 
  

  

  For 
  a 
  circular 
  element 
  of 
  the 
  wave-front, 
  of 
  small 
  diameter 
  

   d, 
  having 
  P 
  for 
  its 
  centre, 
  the 
  primary 
  focal 
  line 
  will 
  be 
  an 
  

  

  arc 
  of 
  the 
  circular 
  axis, 
  of 
  length 
  ^p 
  d 
  ; 
  and 
  the 
  secondary 
  

  

  focal 
  line 
  will 
  be 
  a 
  portion 
  of 
  the 
  axis 
  of 
  revolution, 
  of 
  

  

  AS 
  

   length 
  -r-p 
  d 
  . 
  sec 
  6. 
  If, 
  instead 
  of 
  regarding 
  this 
  absolutely 
  

  

  sharp 
  line 
  as 
  the 
  focal 
  line, 
  we 
  follow 
  the 
  usual 
  convention, 
  

  

  and 
  adopt, 
  as 
  the 
  secondary 
  line, 
  the 
  section 
  of 
  the 
  pencil 
  by 
  

  

  a 
  plane 
  at 
  S 
  perpendicular 
  to 
  PS, 
  its 
  two 
  ends 
  will 
  be 
  blurred, 
  

  

  so 
  that 
  it 
  will 
  resemble 
  a 
  figure 
  of 
  8, 
  and 
  its 
  length 
  will 
  be 
  

  

  AS, 
  

  

  AF 
  (L 
  

  

  At 
  any 
  point 
  T 
  between 
  A 
  and 
  S 
  the 
  breadth 
  of 
  the 
  pencil 
  

  

  AT 
  

   in 
  the 
  plane 
  of 
  the 
  diagram 
  is 
  r^ 
  d. 
  and 
  its 
  breadth 
  perpen- 
  

  

  Al 
  grp 
  L 
  L 
  

  

  dicular 
  to 
  the 
  plane 
  of 
  the 
  diagram 
  is 
  ^p 
  d. 
  The 
  ratio 
  of 
  

  

  ST 
  

   the 
  latter 
  breadth 
  to 
  the 
  former 
  is 
  -7-m 
  multiplied 
  bv 
  the 
  

   AP 
  Al 
  

  

  constant 
  -™ 
  , 
  and 
  is 
  unity 
  when 
  SA 
  is 
  divided 
  internally 
  

  

  and 
  externally 
  in 
  the 
  same 
  ratio. 
  

  

  