Harmonic Analysis. 395 



2. If we define the nth component (0^ say) of a periodic 

 function f{t) of period r as the periodic function which is the 

 sum of those harmonics of f{i) whose orders are «, 'in, 5«, In, 

 etc., then 



2„0,.=/W-/(/+^J+/(/ + 2i.)- .... 



-/(.+-2^1^). (I.) 



For if we represent the expression on the right of the above 

 equation by ^p{t), we find by substituting successively for /, 

 t + TJ2n and Z + t/w in it, that 



Hence \p{t) is an odd periodic function of period rjn, that is to 

 say, if 



/(/) =- «o + 2ap sin/(w/ - 6p), 

 where /> = !, 2, 3, 4, etc., 

 then \p{t) is of the form 



\l/{t) = "^bq sin^«(a)/ - (iq ), 

 where ^=1, 3, 5, 7, 9, etc. 



In evaluating »/'(/) therefore, only those harmonics whose 

 arguments are «w/, ?>n(iit, bnwt, etc., need be considered. Ne- 

 glecting all other harmonics in the differenty functions that make 

 up </'(;'), we find that the remainders in the In terms 



/«• -^(' + ^> ^(' + -2^> *'°- 

 are all equal, and that each remainder is the «th component of 



/(/), hence 



W) = 2nCn- 



3. Ify(/) itself contain only odd harmonics as in the case of 

 alternate current periodic functions, then 



and equation I., §2, reduces to 



nCn=f{t)-f(t+~)Jr . . . +/(^+'^-"^2^> (II-) 

 The operation on/(/) mathematically represented on the right 

 hand side of equations I. or II., is practically performed on 



