400 Mr. 0. Heaviside on Electromagnetic Waves, and the 



the internal H. The disturbance, at the surface, of the primary- 

 wave going both ways is, from t = to 2a/ v, 



The amplitude due to both waves is 



^".(it-n)* (18V) 



fiov \ n 2 a 2 / v 



The outward transfer of energy per second per unit- area 

 at any distance r is EH/47T. In the mth system this is 



EH (f m ) 2 a 2 (vQly ul < , . ' . .. » . « 



— _ \j™j v w j _gL. } u ' sm — (— i W ')cosf nt. \ucos 

 47r 4m(u, v) *r cnr K * 



+ (-iw)sin}n«, (186) 



where m is supposed odd, whilst u and — mw are the real 



functions of nr/v obtained in the same way as (184). The 



mean value of the t function is, by the conjugate property of 



u and iv, equation (114), 



= -n/2v. 



Using this, and integrating (186) over the complete surface 



of radius r, giving 



CCs rf \2 7Q 47rr 2 m(m+l) 



jj( y Q»)' d S= 2m v +1 > • • • (187) 



we find the mean transfer of energy outward per second 

 through any surface enclosing the sphere to be 



m(m. + l) (fmfuy 



2(2m+l) fi v ' ^ 00J 



if (/ TO )vQL cos nt is the vorticity of the impressed force. 



slight change suffices to make (185 d) represent the solution from the first 

 moment of starting the impressed force. Thus, let it start when t=0, 

 and let the f\ in equation (185 d) be (/i) cos nt. Effect the two integrations' 

 thus, 



f-C/Bj*"* ? 4=(/)J(i-cos M o, 



vanishing when £=0, and then operate with the exponentials, and we 

 shall obtain (185 d) thus modified. To the first line must be added 



2fj. vr n 2 ar' 



and to the second line its negative. Thus modified, (185 d) is true from 

 t = 0, understanding that the second line begins when t=(r-a)/v, and the 

 first when t = (r+ a)Jv. The first of (185 a) is therefore true up to distance 

 r—vt— 2a, when this is positive. In the shell of depth 2« beyond, it fails. 



