﻿790 
  Mr. 
  A. 
  Eagle 
  on 
  the 
  Form 
  of 
  the 
  Pulses 
  

  

  In 
  order 
  to 
  prevent 
  having 
  to 
  duplicate 
  such 
  equations, 
  we 
  

  

  cos 
  

   will 
  write 
  them 
  with 
  . 
  , 
  in 
  which 
  either 
  the 
  upper 
  or 
  the 
  

   sm 
  ^^ 
  

  

  lower 
  may 
  be 
  taken 
  ; 
  but 
  in 
  one 
  equation 
  upper 
  must 
  be 
  

  

  taken 
  with 
  upper 
  and 
  lower 
  with 
  lower. 
  

  

  Equations 
  (7) 
  and 
  (8) 
  may 
  very 
  conveniently 
  be 
  expressed 
  

  

  in 
  the 
  form 
  o£ 
  the 
  theorem 
  : 
  

  

  I£ 
  

  

  then 
  

  

  fee 
  

  

  Jo 
  

  

  J 
  *<^'''') 
  1\b 
  "*^ 
  '^* 
  = 
  1'/'^"*^ 
  • 
  

  

  (9) 
  

  

  In 
  this 
  form 
  the 
  equations 
  are 
  useful 
  for 
  finding 
  certain 
  

   definite 
  integrals 
  ; 
  for 
  instance, 
  from 
  the 
  result 
  

  

  I 
  

  

  a 
  

  

  ^ 
  m" 
  -h 
  a' 
  

  

  we 
  can 
  at 
  once 
  deduce 
  

  

  Jo 
  

  

  cos 
  mx 
  7 
  IT 
  -am 
  

  

  da; 
  =-^e 
  

   x^ 
  + 
  a^ 
  2a 
  

  

  Further 
  examples 
  will 
  not 
  be 
  given 
  here, 
  as 
  the 
  author 
  

   hopes 
  to 
  discuss 
  some 
  results 
  which 
  may 
  be 
  obtained 
  from 
  

   equations 
  (9) 
  in 
  another 
  paper. 
  

  

  Equations 
  (7) 
  and 
  (8) 
  enable 
  us 
  to 
  solve 
  at 
  once 
  (3) 
  

   and 
  (4). 
  Multiplying 
  each 
  side 
  of 
  (3) 
  by 
  cos 
  ax 
  da, 
  and 
  

   integrating 
  from 
  to 
  cc 
  , 
  we 
  obtain 
  by 
  (7) 
  

  

  f(x) 
  = 
  ^\ 
  F 
  (ay 
  COS 
  ux 
  da, 
  .... 
  (10) 
  

  

  "^ 
  Jo 
  

  

  which 
  gives 
  the 
  form 
  of 
  the 
  even 
  pulse 
  ; 
  similarly, 
  the 
  odd 
  

   one 
  is 
  given 
  by 
  

  

  f(x) 
  = 
  - 
  r 
  F(a)* 
  sin 
  ax 
  da 
  (11) 
  

  

  •^ 
  Jo 
  

  

  We 
  will 
  now 
  determine 
  the 
  form 
  of 
  the 
  pulses 
  for 
  some 
  of 
  

   the 
  different 
  formulae 
  that 
  have 
  been 
  proposed 
  for 
  representing 
  

   the 
  distribution 
  of 
  energy 
  in 
  full 
  radiation. 
  

  

  Both 
  Lord 
  Rayleigh's 
  and 
  Wien's 
  formulae 
  are 
  included 
  

   under 
  

  

  \d\ 
  = 
  Ce''\''-^e'^od\, 
  

  

  the 
  former 
  being 
  obtained 
  by 
  putting 
  ?i 
  = 
  l 
  and 
  the 
  latter 
  by 
  

  

  