74 



PROCEEDINGS OP SECTION A. 



arid in either of these formulje the trio;onometrical expressions in 

 Section 9 for the vectors can be substituted. In the first, however, it 

 must be noted that both a,^ _ i and a,, + i are to be taken of order q 

 (odd). Thus the fundamental harmonic of E is — 



_ wmr] \ ^.^ ^^ _^ ^^^ + b, + B.) I 



from the first expression, or — 



= -^ D. si,, (,^ + i_ _y;) 

 P «i 



from the second, t, being equal to J), l ~'^' . 



Similarly the total alternating e.m.f. II generated in the field 

 circuit is given by either — 



H = - '"^\ "S;>(a,_i + a, + ,), • 



9 



or- 



2 



so that the fundamental harmonic of H is equal to either — 



^^^'^ r ^ sin {2^t + J^ + ^j + J- sin (2u,t + ^. + ^, + J + ^) 1 



2^^ --^= sin {2..t + b,+/3.-4>. + -rr). 



p Sj(Tz 



11. The mean value of the product sin (awf + 6) sin (Ixat + ^) 

 being zero when a and S are unequal, and | cos {6 — </>) when a and h 

 are equal, we find that the mean value of x'' where ^ = 2 a, is — 



and the mean value of ^ '^ , where $ = ~- + 2 a^ is — 



1 2 



4 2 "^' ■ 



Again, for the same reason, if a and (3 be any two vectors representing 

 harmonics of the same order, and if V a /? be the product of the lengths 

 of a and (i into the sine of the angle from a to yS measured in the 

 positive direction, then the mean value of the product — 



TT 



I'^a. into (i 

 = 1 Vtty8= -iVySa. 



Applying these principles to determine the mean value E^r of the 

 product of E and oc, that is, of the electrical power developed in the 



