504 Displacement Currents [OH. xvn 



General solution (Kirchhoff). 



580. A still more general form of solution has been given by Kirchhoff. 

 Let <E> and ^ be any two independent solutions of the original equation, so 

 that 



dt* ~ 

 By Green's Theorem (equation (101)) 



_ I! ( ^ - ^ 



by equations (542). The volume integrations extend through the interior 

 of any space bounded by the closed surfaces S lt 8 2 , ..., and the normals to 

 8 lt $ 2 , are drawn, as usual, into the space. If we integrate the equation 

 just obtained throughout the interval of time from t = t' to t = + t", we 

 obtain 



(543). 



So far ^ has denoted any solution of the differential equation. Let us 

 now take it to be - F (r + at), this being a solution (cf. equation (536)) what- 

 ever function is denoted by F, and let F (x) be a function of x such that it 

 and all its differential coefficients vanish for all values of x except x = 0, while 



/: 



Such a function, for instance, is F(x)=* Lt j-\ ^ . 



c=0 * (* + c ) 



We can choose if so that, for all values of r considered, the value of 

 r - at' is negative. The value of r + at" is positive if t" is positive. Thus 

 F (r + at) and all its differential coefficients vanish at the instants t = t" and 

 t = t', so that the right-hand member of equation (543) vanishes, and the 

 equation becomes 



Let us now suppose the surfaces over which this integral is taken to be 

 two in number. First, a sphere of infinitesimal radius r , surrounding the 

 origin, which will be denoted by S lt and second, a surface, as yet unspecified, 



