Practical  Method  of  Harmonic  Analysis.  27 
are  all  equal,  and  .that  each  remainder  is  the  nth  component 
oiftt)  ;  hence 
f(t)  =  2nCH. 
3.  If  f(t)  itself  contain  only  odd  harmonics  as  in  the  case 
of  alternate-current  periodic  functions,  then 
f(t)=-f(t+z2), 
and  equation  I.,  §  2,  reduces  to 
»c.=/(0-/(*+£)+  •  •  •  •  +f(t+,T^l'L}   ■  ("•■) 
The  operation  on  f(t)  mathematically  represented  on  the 
right-hand  side  of  equations  I.  or  II.  is  practically  performed 
on  alternate-current  waves  by  the  wave-tracer  and  analyser  * 
designed  by  the  author.  In  the  simplest  case,  when  n  =  l, 
the  wave-tracer  gives  the  first  component  of  the  periodic 
quantity  operated  on,  which  in  the  case  of  alternating  electric 
currents  is  the  full  wave.  By  the  movement  of  two  pairs  of 
brushes  n  can  be  made  3,  or  5,  or  7,  in  which  cases  the 
analyser  will  give  the  3rd,  5th,  or  7th  components  of  the 
wave  respectively. 
Now,  in  practical  investigations  with  this  apparatus  on 
alternating-current  waves  whose  harmonic  expressions  were 
required,  it  was  found  much  better  to  obtain  by  its  means 
only  the  full  wave-trace,  and  then  by  an  arithmetical  process 
identical  with  the  action  of  the  analyser  and  "indicated  by 
equation  II.  above,  to  obtain  the  3rd  and  higher  components 
of  the  wave,  and  thence  to  deduce  its  harmonics. 
This  method  of  harmonic  analysis  was  drawn  attention  to 
in  the  paper  already  quoted,  and  though  based  on  a  different 
formula  to  that  of  Wedmoref,  is  practically  similar  to  his. 
It  is  more  suitable,  however,  for  waves  containing  only  odd 
harmonics;  and  as  I  have  had  considerable  experience  in  its 
use  during  the  last  two  years  and  have  found  it  both  expe- 
ditious and  accurate,  it  is  possible  that  a  short  account  may 
be  of  value  to  those  interested  in  alternating-current  work. 
4.  In  wave  graphs  it  is  more   convenient  to  use  angular 
abscissae  x  where  .     n    ,» 
,r  =  cot  =  Z7rtjr. 
Making  this  substitution  in  the  equation  y=f(f),  it  becomes 
y=g(x)  say,  where 
*  Lyle,  "  Preliminary  Account  of  a  Wave-Tracer  and  Analyser," 
Mag.,  Nov.  L903. 
t  Wedmore,  'Journal  [nst.  Elect.  Engineers,'  vol.  kxv.  p.  _-l  (1896). 
