﻿620 
  Mr. 
  A. 
  Stephenson 
  on 
  Periodic 
  

  

  2f(t) 
  can 
  be 
  split 
  up 
  into 
  the 
  even 
  function 
  f{t)+f{— 
  t), 
  

   and 
  the 
  odd 
  function 
  f(t)—f( 
  — 
  t); 
  and 
  these 
  are 
  expressible 
  

   in 
  cosine 
  and 
  sine 
  series 
  respectively. 
  Hence 
  the 
  equation 
  

   of 
  motion 
  is 
  

  

  00 
  

  

  as 
  + 
  {X 
  -f- 
  2n 
  2 
  % 
  {oLf 
  cos 
  rnt 
  + 
  j3 
  r 
  sin 
  rnt)}w 
  = 
  0, 
  

   i 
  

  

  where 
  n 
  is 
  the 
  frequency 
  of 
  variation 
  per 
  2ir 
  units 
  of 
  time. 
  

   The 
  solution 
  is 
  

  

  00 
  

  

  x= 
  % 
  {A 
  r 
  sin 
  (c— 
  rn) 
  t 
  + 
  B 
  r 
  cos(c 
  — 
  rn)t}, 
  

   where 
  -co 
  

  

  A,{X-(c-7T0 
  2 
  } 
  + 
  n 
  2 
  X{« 
  s 
  (A,_ 
  s 
  H-A 
  r+s 
  )-A(B 
  r 
  _ 
  s 
  -B 
  r+s 
  )} 
  = 
  

  

  (r)> 
  

  

  These 
  equations 
  determine 
  the 
  A's 
  and 
  B's 
  in 
  terms 
  of 
  

   A 
  and 
  B 
  , 
  and 
  the 
  eliminant 
  gives 
  c. 
  Under 
  the 
  same 
  

   conditions 
  as 
  before 
  

  

  A 
  r 
  =^(« 
  r 
  A 
  -/3rB 
  ), 
  

  

  A_ 
  r 
  = 
  T2 
  (a 
  r 
  A 
  -}-/3 
  r 
  B 
  j, 
  

   B 
  r 
  =^(«,Bo 
  + 
  /5rA 
  ), 
  

   B- 
  r 
  = 
  -, 
  {u 
  r 
  B 
  — 
  /3,A 
  ), 
  

  

  when 
  terms 
  above 
  the 
  first 
  degree 
  in 
  the 
  a's 
  and 
  /3's 
  are 
  

   neglected. 
  

  

  A 
  r 
  + 
  A_ 
  r 
  = 
  2 
  — 
  , 
  A 
  , 
  

   B 
  r 
  — 
  B_ 
  r 
  =2 
  -g 
  A 
  , 
  

  

  B 
  r 
  +B_, 
  =2-5B 
  . 
  

  

  r 
  

  

  