Forced Vibrations of Electromagnetic Systems. 495 

 The results are 



-A 3 =J 0r .-<W--a 2 )*, (385) 



v 

 2 T \a? 



A *=&^ {vH2 - r2) ~ K ■ • • (386) 



From these derive E 3 and E 4 by time-differentiation, and H 3 , 

 H 4 by space-differentiation, according to 



1 rJA 



curl A=/*H, orH=-~. . . . (387) 



We see that the value of E 3 at the axis, say E , is 



E =evt(vH 2 -a 2 )-i; (388) 



and by performing the operation J 0r in (385) we produce, if 



a _ e r r2 1 1 vH \ ^ ( 1 Qv2t2 5v ^\ 



45r 6 / 1 15vH 2 35^ 4 21vH\ , -i . 



from which we derive 



in.?[i + ^+3S<w + *r> 



+ g.kw (5 " 4 + 20oW+ 8c¥ ) +•••]' < 39 °) 



These formula commence to operate when vt=a at the 

 axis and when vt = a + r at any point r<a, and continue in 

 operation for ever after. 



72. Lastly, perform the operation (2/sa)J lrt in (386), and 

 we obtain 



^"2»L« »( u 3 u 5 J 64 U 5 ™ 7 ^ 9 / 



45a 6 (_ 5 135^ _ 315^ 4 231wV\ . -i /Qfn . 



from which we derive 

 E * = lf [ 1+ 5?(** , +**> +|J (8^ + 40,W + 15^) 



+ 



45a 6 ' 



4.36.64m 18 



(112f¥ + 1176^V + 1470w 2 « s r 4 + 245r 6 ) + . . .](392) 



