544 Prof. Arthur Schuster on 



can be simplified if the only part of the boundary at which </> 

 does not vanish consists of a plane surface. Write 



so 



that 



2<t> = fa + fa. • (8) 



Both yfr x and fa satisfy separately the differential equation 



The value of fa at a given point and time is expressed as 

 the potential due to a double layer. Of such a potential it is 

 known that it shows a discontinuity at the surface equal to 

 2</> (£) . Hence if the surface is plane the fa at the surface must 

 be equal to 4>(t). The function fa takes, therefore, the same 

 value as cj) a ^ the surface, and from (8) it follows that the 

 same must hold for yfr 2 also. 



The value of -\fr 2 is expressed as a surface potential, and the 

 form of the integral shows that at the plane surface 



<^h _ /rA _ # 

 dN ~ JK -> dN' 



Hence the differential coefficient of fa at the surface is the 

 same as that of </>, and it follows again from (8) that the same 

 must also hold for fa. 



Both fa and fa satisfy, therefore, all conditions : hence, 

 taking as axis of z the direction of the normal, we may, for 

 the special case under consideration, write Kirchhoff's equation 



= L f A "/.as, 



dzj r 



all of which forms may be useful. Taking the first, perform- 

 ing the differentiation, and rejecting the terms which vary 



