HARMONIC ANALYSIS. 85 



The values of the coefficients d, a^, as, . . . &., b,, h, . . . are given by 



C=yy.dt (8) 



r 

 a. = J, fy -cos (i ^t).dt (9) 



h^liy.sm{ip).dt (10) 



{i=\,2, 3, . . . ) 



By using a sufficient number of terms any function whatever (straight 

 line, irregular curve) of t within the period T can be represented by the 

 series (7). The resolution into such a series is termed "harmonic analy- 

 sis." The results of the analysis are applicable only to the curve within 

 the period analyzed, that is, within T, unless the curve repeats itself 

 exactly outside of T. 



To obtain a more convenient form for (7) it is sufficient to put 



a= — c.sin 9, h = c.Q,os q, 



that is, (11) 



c = v^oMT', tang=~p- 



whence we have 



2/ = C + c..sin(|'«-g,)+c..sin(^<-g.)+C3.8in(|^f-g3)+ . . . (12) 



It is often convenient to use the form (6), whereby the equations 

 (7,8,9, 10) become 



i/ = C + a,.cosA;< +a2.cos2A;< + Oa.cos 3fci-|- . . . (13) 



-f 6,. sin kt + 62- sin 2kt + 63. sin Zkt-\- . . . 



3ir 



C=lfy.dt (14) 



2ti 



