﻿198 
  Mr. 
  James 
  Walker 
  

  

  on 
  

  

  and 
  the 
  maximum 
  and 
  minimum 
  values 
  o£ 
  y 
  are 
  y=^aXf, 
  

   occurrinor 
  when 
  

  

  o 
  

  

  X 
  = 
  x,n 
  = 
  (b± 
  \/ 
  6^ 
  + 
  a^c) 
  l(ac) 
  . 
  

  

  3. 
  First 
  considering 
  sin 
  27, 
  we 
  see 
  that 
  sin 
  27=0, 
  or 
  

   the 
  light 
  is 
  plane 
  polarized, 
  when 
  tanR=0 
  and 
  when 
  

   tan 
  R= 
  — 
  tan 
  2a, 
  the 
  corresponding 
  values 
  of 
  tan2(^ 
  — 
  a) 
  

   l)eing 
  and 
  — 
  tan 
  4a. 
  Sin 
  27 
  attains 
  its 
  maximum 
  and 
  

   minimum 
  values 
  sin 
  2a 
  cot 
  2y8tan 
  r^, 
  when 
  

  

  ^ 
  -D 
  ^ 
  I 
  + 
  a/H- 
  tan2 
  2a/ 
  sin2 
  2.6 
  . 
  

  

  tan 
  K= 
  tan 
  r,n= 
  — 
  '-^ 
  — 
  ^tT'-^L/d 
  ' 
  

  

  tan 
  2 
  a/ 
  sm^ 
  2p 
  

  

  writing 
  tan 
  2a/ 
  sin 
  2^ 
  = 
  t^n 
  yjr, 
  yjr 
  being 
  a 
  positive 
  angle 
  less 
  

   than 
  TT, 
  

  

  tan 
  rm 
  = 
  sin 
  2/9 
  cot 
  '^/2, 
  or 
  — 
  sin 
  2/S 
  tan 
  ylr/2 
  , 
  

  

  and 
  since 
  tan 
  R= 
  sin 
  2/3 
  tan 
  S/2, 
  the 
  maximum 
  and 
  minimum 
  

   values 
  o£ 
  sin 
  27 
  occur 
  when 
  B 
  = 
  n7r—'\jr. 
  

  

  The 
  values 
  o£ 
  tan 
  R 
  corresponding 
  to 
  the 
  maximum 
  and 
  

   minimum 
  values 
  of 
  sin 
  27 
  may 
  be 
  written 
  

  

  tan 
  E; 
  = 
  sin 
  2fi 
  . 
  x, 
  where 
  x 
  = 
  cot 
  '\jrl2 
  or 
  — 
  tan 
  yjrl2, 
  

  

  ^nd 
  tan 
  2a 
  = 
  - 
  2 
  sin 
  2fi 
  cot 
  '^/2/(l 
  - 
  cot^ 
  ^fr/2) 
  

  

  = 
  2 
  sin 
  j3 
  tan 
  i/r/2/ 
  (1 
  - 
  tan^ 
  'xlr/2) 
  

   = 
  -2sm2^xl(l-x^); 
  

  

  substituting 
  these 
  values 
  in 
  the 
  expression 
  for 
  fan2(^— 
  a), 
  

   we 
  find 
  

  

  tan 
  9{f) 
  r.\-9^h. 
  9 
  R 
  <l 
  + 
  ^t-')(l 
  - 
  COS 
  4/3 
  . 
  A'^) 
  ^ 
  

   tan 
  2(^-a) 
  _2 
  sm 
  2 
  /3^-j--,^^^-^,^^^__ 
  ^^^^^ 
  ^ 
  ^^,^ 
  , 
  

  

  and 
  unless 
  ^2= 
  sec 
  4/3 
  

  

  tan 
  2(^-a) 
  = 
  2 
  sin 
  2/3 
  j^-^^ 
  = 
  - 
  tan 
  2a. 
  

  

  J- 
  X 
  

  

  But 
  when 
  x^ 
  = 
  iiec4:j3 
  

  

  tan 
  2a=ycot2 
  2/3-1, 
  or 
  tan^ 
  2^8= 
  cos^ 
  2ci 
  ; 
  

  

  hence 
  corresponding 
  to 
  the 
  maximum 
  and 
  minimum 
  

   values 
  of 
  sin 
  27, 
  we 
  have 
  tan 
  2 
  (^ 
  — 
  a) 
  = 
  — 
  tan 
  2a, 
  unless 
  

   cos 
  2a 
  = 
  + 
  tan 
  2/3, 
  a 
  case 
  that 
  will 
  be 
  considered 
  later. 
  

   Further, 
  sin 
  27= 
  ±1, 
  only 
  if 
  

  

  tan 
  '\jrl2 
  = 
  sin 
  2a 
  cos 
  2/3, 
  or 
  cot 
  yjr/2 
  = 
  sin 
  2a 
  cos 
  2/3, 
  

  

  that 
  is 
  if 
  cos 
  2« 
  = 
  ib 
  tan 
  2y8, 
  and 
  consequently, 
  except 
  in 
  these 
  

   cases, 
  6 
  determines 
  the 
  plane 
  of 
  maximum 
  polarization. 
  

   Hence, 
  reserving 
  for 
  future 
  consideration 
  the 
  cases 
  in 
  

  

  