422 ANALYSIS OF CURVES. 



or 5, <tc., times the periodicity of the fundamental. That 

 this must be so follows from the consideration that any 

 two points on the curve which are exactly half a period 

 apart must be equal values of the varying quantity, but 

 having opposite signs, since they are values of the 

 electromotive force generated at similar positions under 

 opposite poles. Tfiis is only true in the case of 

 curves having a frequency which is an odd multiple of 

 that of the fundamental. Thus, on referring to Fig. 199, 

 if the distance between the two vertical lines AB is one 

 full period of the fundamental, the frequency of the curve 

 will be two per cycle of the main curve, so that the 

 curve is the second (i.e., an even) harmonic. Taking any 

 two vertical lines, such as those shown dotted at a, b, 



FIG. 201. THIRD HARMONIC IN PHASE. WAVK FLATTENED. 



half a period apart, it is evident that the values separated 

 by half a period are equal, but similar in sign. A curve 

 having this as one of its harmonics would, consequently, 

 not be symmetrical in the sense of its having its negative 

 half wave an exact repetition of its positive half wave, 

 although it would be symmetrical in the sense that each 

 wave would be similar to the-preceding one. 



The expression for the harmonics of a symmetrical 

 wave form obtained from an alternator may consequently 

 be written in the following form, in which the even 

 harmonics are omitted : 



e = E sin (2 irnt + 0) + E 8 sin (6 irnt + 3 ) + 

 -fffi (10 irn .+ S ) + . . . 



A curve which contains only even harmonics will be 

 symmetrical in a different sense from the curve containing 

 only odd harmonics. Each half-wave wave will be similar 



