DETERMINATION OF WAVE FORM 663 



.k . 

 p , andp is not a whole number, both the series in (21) 



reduce to zero. 

 Consequently 



[Y P ]i + [Fp], + [F P ], + . . . + [Fp]p = (25) 



1c 



That is, when p is not a whole number the sum of P equally 



spaced ordinates is zero. This is the second of the two laws upon 

 which this method depends. 



These relations are used as follows: The wave is plotted and 

 any point is taken as the origin. 



At the origin, t = 0, all the sine terms are zero and all the cosine 

 terms have their maximum values, that is, B i} B Z} B$ . . .so 

 Y l = Bi + B, + 5 5 + . . . 



To find B z : 



Between a and b there are three complete periods of the third 

 harmonic, nine complete periods of the ninth harmonic, fifteen 

 complete periods of the fifteenth harmonic, and so on. 



Divide the base ab into three equal parts. Then P = 3 and 



k 



p = 1 f or the third harmonic 



k 



p- = 3 for the ninth harmonic 



k 

 - = 5 for the fifteenth harmonic. 



These are all whole numbers and by (22) 



[F 3 ]i + [F 3 ] 2 + [F,], = 3[ 3 + B, + Bis + 21 + ] 



To find B 6 : 



Divide the base ab into five equal parts, P = 5; between a and 

 b there are five complete periods of the fifth harmonic, fifteen 

 of the fifteenth harmonic, so 



^ = 1 for the fifth harmonic 



= 3 for the fifteenth harmonic, 



Consequently 1 



[F 6 ]i -f [F 6 ] 2 -f [FJa + [F B ] 4 + [F 6 ]6 = 5[ 6 + B u + B+ ... J 



