26  Prof.  T.  R.  Lyle  on  an  Expeditious 
The  number  p  is  called  the  order  o£  the  harmonic,  ap  its 
amplitude,  and  6   its  phase. 
I£,  in  addition,  f(t)  be  such  that 
/(0  =  -/(*+t/»), 
then  it  is  easy  to  see,  by  substituting  t  +  r/2  for  t,  i.  e.  cot  +  ir 
for  cot  in 
y  =  a0  +  2ap  sin  p  (cot -dp), 
that  in  order  for  vt  to  be  =  —yt+T/2, 
aQ  =  0,    a2  =  0,    a4=0,  &c. 
Hence  in  this  case  the  constant  term  vanishes  and  the 
harmonics,  of  which  f(t)  is  the  sum,  are  all  of  odd  order. 
When  such  is  the  case,  f(t)  is  called  an  odd  periodic 
function.  This  is  the  type  generally  met  with  in  alternating 
electric-current  investigations. 
2.  If  we  define  the  nth.  component  (Cn  say)  of  a  periodic 
function  f(t)  of  period  r  as  the  periodic  function  which  is 
the  sum  of  those  harmonics  of  f(t)  whose  orders  are 
n,  3 n,  5n,  In,  &c,  then 
«-o.a-/(o-/(t+i)+/(*+*i)-.-.- 
-/(t+S=l£) (I.) 
For  if  we  represent  the  expression  on  the  right  of  the 
above  equation  by  ^jr(t),  we  find  by  substituting  successively 
for  t,  t  +  rj2n,  and  t  +  r/n  in  it5  that 
^)=-y(*+^)=f(*+9- 
Hence  ^fr(t)  is  an  odd  periodic  function  of  period  r/n,  that 
is  to  say,  if 
f(t)=a0  —  %ap  sm  p  (at— 0p% 
where  p=l,  2,  3,  4,  &c,  then  yfr(t)  is  of  the  form 
ir(t)  =  Zbq  sin  qn  (cot -@q), 
where  q  =  l,  3,  5,  7,  9,  &c. 
In  evaluating  yfr(t')1  therefore,  only  those  harmonics  whose 
arguments  are  ncot,  2>ncot,  Snout,  &c,  need  be  considered. 
Neglecting  all  other  harmonics  in  the  different  /  functions 
that  make  up  ty(t),  we  find  that  the  remainders  in  the 
2?i  terms 
m,  -/(*+£)    /(«+»£) 
&c, 
