M 1+ i5M x ] ; 



2 heory of the Optical Properties of Metals. 439 



The applications of this function in the present paper are 



to be made with X periodic — say, of period — ; we may 

 then use ^ 



X=X</K 



It is then easily found that 



ip'i 



or, since T m = l m u > 



we have /«/, |l + ***' ^A . 



\ i + ^K^ 5 / 



The ' current of electricity in the metal in this case is 

 directed entirely along the axis of x and is of a density 



r+» /"•+» r»+<» 

 J*= e&dZdvdt 



%} — oo %) — oo «y— oo 



3 V ttO-oo J-oo J-oo 1+tpr^ 



This integral may be transformed by the usual spherical 

 polar transformation, and thereby reduces at once to 



T Sir^eqLX /f [ u s e-^du 



^ l+ipl m u s 

 If we now use 



2_1 j 



_ 2 OS - — P m 



this integral becomes 



</ s J 



3«> « J.i + IM 



1 > 



3(? s ^o !+(£*)' 



and it is not possible to reduce it further, except perhaps to 

 another irreducible integral of the integral-logarithmic type. 



