512 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



If we perform upon the equations (13) the operation of curl, 

 which is typified by the result of differentiating the third of 

 equations (B) by y and subtracting it from the second differentiated 



by z we obtain, after adding and subtracting 



u 



. s (m m\ . d ftx ar 



A 3i(^-^) = * X -S- X (te + S y 



Now supposing the medium to be homogeneous, that is e 

 and /-t constant, making use of the equations (C) and (D), and 

 supposing there is originally no electrification, we have 



+?!+*. 



da dy dz 

 and making use of the first of equations (A) we transform ( I ) into 



Proceeding in like manner we obtain for the other components, 



(2) 





We thus find that in insulators each component of the two 

 fields satisfies a differential equation of the form 



(3) ?*-*A*, , . 



where a I/ A *J pe. 



Since this is an equation of great importance in mathematical 

 physics, we shall investigate its general solution. Let us multiply 

 both sides of the equation by the element of volume dr and 

 integrate throughout the volume bounded by a closed surface 

 $, applying the divergence theorem to the right-hand member, 



