Forced Vibrations of Electromagnetic Systems. 141 



where all the J's operate on st*J — 1; thus, e.g. (Bessel's), 



J 3 =J 3 (st V^i)- 



But a much better form than (52), suitable for calculating 

 the shape of the wave speedily, especially at its start, may be 

 got by arranging in powers of z — vt, thus 



H=|H 0f -,{ 1 + ^(1-1) + |/ 2 (l-£/ 



true when z<vt, where / u / 2 , &c. are functions of t only, of 

 which the first five are given by 



/ s =f(i + f> 



At the origin, H is given by 



H=iH e- 2 ^ (54) 



and is therefore permanently ^H when # = 0. At the front 

 of the wave, where z = vt, 



H = iH e-** (55) 



Now, to represent the E solution corresponding to (51), we 

 have only to turn H to E in (53), and change the sign of s 

 throughout, i, e. explicit, and in the /'s. Similarly in (52). 

 Thus, at the origin, 



E=p o e-^ (56) 



and at the front of the wave 



E=iE e-* (57) 



9. Again, let H = ^H on the left side, and H=— iH on 

 the right side of the origin, initially. The E that results from 

 each of them is the same, and is half that of (49) ; so that 

 (49) still expresses the E solution. This case corresponds to 

 an initial electric current of surface-density H /47r on the 



