(28 - ^) II S sin (e - |) . (3) 



248 ALTERNATING CURRENTS 



or 



since sin 2 (0 ^ J = ^l cos 2^0 ^.Jj. The upper sign in the 



last term corresponds to < 0, and the lower sign to > 0. 



To obtain the mean rate of heat production at P, we must find 

 the mean value of i 2 during a half-period, that is, between = and 

 = TT. Now, we notice that the first two terms in (3), |I 2 + ^I N 2 > 

 are constant ; their mean value is thus I 2 -J- JI N 2 . The third term 



contains the factor cos (20 -~\ which is of frequency double that 



of the supply current ; thus half a period of the supply current 

 from = to = TT will correspond to a whole period of the 



term cos (29 ^J, and since the mean value of the cosine over a 

 whole period vanishes, this term drops out in the mean value of i 2 . 

 Lastly, we have the term II N sin (0 ^ j, the plus sign to be taken 



so long as < <f>, and the minus sign when > 0. If we plot this 

 term as a function of 0, the area enclosed by its graph is 



II N { j sin (0 - )dO - J sin (0 - 



= - 21I N cos - 



and the mean ordinate of the curve, obtained by dividing the area 

 by the base, is 



- ?II N cos (* - 



We thus obtain for the mean value of the square of the current 

 at P 



cos - 



This expression shows that the heating is greatest at E and S, the 

 points of slip-ring connection, and least at T, the point halfway 

 between them. 



If, using (2), we plot the expression (4) as a function of for 

 N = 2, 3, 4, 6, and 12, the extreme values of being always and 



-^, so as to correspond to the distance between two neighbouring 

 slip*rings, we obtain the curves of Fig. 153. The value of I the 



