﻿H. 
  8. 
  Uhler 
  — 
  Note 
  on 
  Paper 
  by 
  Ch. 
  Fabry. 
  145 
  

  

  .N^ct+Zabs 
  +(2ac+6») 
  s 
  2 
  +2 
  (ae+bc) 
  s 
  3 
  + 
  (2a/+26e+c 
  2 
  ) 
  s 
  4 
  + 
  

   2(a$H-&/+ce)« 
  6 
  (4) 
  

  

  N 
  i 
  =:a 
  4 
  + 
  4a 
  t 
  bs+%a* 
  (2ac-f-36 
  2 
  ) 
  s 
  2 
  +4a 
  (« 
  2 
  e-f 
  3a6c~f-6 
  3 
  ) 
  «■+ 
  

   (4ay+6aV+12a 
  2 
  6e+12a£ 
  2 
  c+6 
  4 
  ) 
  * 
  4 
  -f 
  4 
  (a^+ 
  

  

  3a"ce+3a'ft/+3aAc 
  a 
  +3aJ 
  ,, 
  e 
  + 
  ft 
  a 
  c) 
  s 
  5 
  (5) 
  

  

  Substituting 
  the 
  expressions 
  for 
  N, 
  N 
  2 
  , 
  and 
  N 
  4 
  , 
  given 
  

   respectively 
  by 
  equations 
  (3), 
  (4), 
  and 
  (5), 
  in 
  equation 
  

   (2), 
  and 
  equating 
  to 
  zero 
  the 
  collected 
  coefficients 
  of 
  the 
  

   powers 
  0, 
  1, 
  2, 
  3, 
  4, 
  and 
  5 
  of 
  s, 
  it 
  will 
  be 
  found 
  that 
  

  

  ' 
  e 
  = 
  

  

  4>i 
  2 
  sin 
  J. 
  

  

  6= 
  - 
  '£ 
  v 
  cot 
  £4, 
  y 
  4 
  (15 
  + 
  24cos^4-7cosM) 
  

  

  7 
  ~ 
  128?2 
  3 
  sin 
  2 
  J. 
  _ 
  , 
  

  

  _ 
  v 
  2 
  _ 
  v 
  (sinM+2 
  cos 
  ^4) 
  

  

  C 
  ~~ 
  8n' 
  ^ 
  _ 
  32^ 
  4 
  sin~Z^in 
  2 
  ^ 
  ' 
  

  

  Replacing 
  these 
  expressions, 
  together 
  with 
  s 
  = 
  sin 
  D, 
  

   in 
  equation 
  (3) 
  the 
  final 
  general 
  formula 
  is 
  obtained. 
  

  

  In 
  particular, 
  if 
  ^4 
  = 
  90° 
  and 
  v=l, 
  the 
  preceding 
  

   coefficients 
  reduce 
  to 
  those 
  given 
  in 
  Fabry's 
  formula 
  (3) 
  

   and 
  to 
  his 
  very 
  special 
  cases 
  (1) 
  and 
  (2). 
  [Fabry's 
  

   formula 
  (3) 
  ends 
  with 
  sin 
  4 
  D.] 
  

  

  A 
  very 
  convenient 
  alternative 
  series, 
  which 
  converges 
  

   rapidly, 
  is 
  given 
  below. 
  

  

  T 
  = 
  tan-J-^4, 
  t 
  = 
  l&n^J), 
  

  

  jy=n 
  + 
  a 
  t 
  + 
  bt*+ 
  ct 
  3 
  + 
  At* 
  + 
  e* 
  5 
  + 
  it\ 
  

  

  V 
  

  

  a 
  = 
  — 
  -, 
  

  

  T 
  

  

  Zn 
  

  

  v 
  

  

  c= 
  M* 
  + 
  »• 
  + 
  **), 
  

  

  on, 
  r 
  

  

  e 
  = 
  -£-,[v 
  4 
  - 
  (« 
  4 
  + 
  3*V 
  -2v 
  4 
  ) 
  r 
  2 
  - 
  v 
  2 
  (3w 
  9 
  + 
  v 
  u 
  ) 
  r 
  4 
  ], 
  

  

  

  f 
  = 
  v 
  5 
  4 
  [16v 
  4 
  - 
  8v 
  2 
  (8w> 
  - 
  5v 
  2 
  ) 
  r 
  2 
  + 
  (24;i 
  4 
  - 
  88vV 
  -43v> 
  4 
  -f- 
  

   4v 
  9 
  (8ra 
  3 
  - 
  7v 
  2 
  )r 
  6 
  ]. 
  

  

  Sloane 
  Physical 
  Laboratory, 
  Yale 
  University, 
  

   New 
  Haven, 
  Conn., 
  December, 
  1919. 
  

  

  