HARMONIC ANALYSIS 



391 



For when n 

 2 



1, 



when n 

 2 



2, 



47* 



m 

 STC 



y m . lC os 



2(m - 





2 ( 47C STC 



Bo - { w + Vi cos -- (- 1/2 cos -- h 

 m \ Jl m m 



If 2 " 



(3) Now A n - - y sin n6 dO 



"Jo 



4(ra- l)7tn 

 S/m-iCos M j 



m 



A^ is evidently twice the average ordinate of the curve obtained 

 by plotting horizontally and y sin n6 vertically. 



2 



Then A_ = {sum of the ordinates of the 0, y sin n0 curve} 

 1 



= -y sin +yi sin 



sin 





y m _, sin 



2n(m 



and this will give the coefficient of any sine term in the resulting 

 series. 



It follows, therefore, that in the operation necessary to obtain 

 any coefficient of the form A,,, the ordinates y , y lt y 2 . . . must 



be multiplied by the sines of the corresponding angles 0, -- , 



Tft 



CttTT 



. , ,, , . 



- , - , etc., and twice the average of the sum of these 

 m m 



products taken. While to get B n the ordinates must be multi- 

 plied by the cosines of the corresponding angles and twice the 

 average of the sum of the products taken. 



If a = - - and from a fixed point radial lines are drawn making 



VYl 



angles 0, a, 2a, 3a, etc., to the horizontal (Fig. 127). 



7 

 -_- - 



FIG. 127. 



