Equation of Electrical Eloiv, 431 



0+Y 



ae tan 0sm 0-fy. 

 The other by 



be tan s in0-= 7. 

 The product is 



or 



ab.e tan ^siD + y sin 0—y, 

 or 



20 



a6 . e tan/3 (sin 2 0— sin 2 7). 

 This quantity, multiplied into an element of time dt, has to 

 be integrated through one period. Since -r- = — , the 



(lit 1% 



integral 1 becomes 



nht C 2d 



°^ e't^Tp (cos 2 7 - cos 20) dO, 



nht C 2 nht C 20 



|i 2 e -*rp cos 2yde-^\ e'^fi cos 20 dd. 



The first term of the integral is 



ahU rt _ - 2e 



— 5 — cos 2<y tan pe ta * . 



The second term is 



tt 2 • n /-> ■ o/i tan/3 



-g^rsm/3cos/3 + 20e p > 

 and therefore the integral is expressed :— 



\= a ^h e ~^ {sin/3cos i g + 20--tan / ecos27}. 



This expression has to be taken between limits. If we con- 

 template one revolution only the limits will be ©! + 27rand X , 

 and the Definite Integral becomes 



obu • _??!_/ __±L\ 



-g^f { — sin /3 cos 8 + 20i + tan /3 cos 2 7 }<? ~ w( 1 — £ tan^ J. 



If the limits are infinity and X the integral becomes 



abu 



_ *" / i 



20, 



-g^- {tan ^ cos 2 7 — sin /S cos ^ + 20!}^ tan ? . («) 



Either of these expressions becomes zero when 



tan /3 cos 2 7 = sin ft cos /3 + 20 j 

 2G2 



