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



Inserting these and the corresponding values in the integrand, 

 and replacing the time derivatives by the curl-components from 

 equations (A) and (B), we have 



dL 



dz ( Ar 



r -- o- 1 *f -w 1 3 -- o-* 



Now integrating by the theorem of 226 (4), this becomes 

 (3) d + H 



+ (XM-YL)cos(nz)}dS. 



We have supposed that there are no intrinsic electromotive forces, 

 but if there are, the components X, Y, Z in (i) must be replaced 

 by X-X', Y-Y', Z-Z' (243), except in the last integral 

 representing the dissipativity, consequently that integral will not 

 be entirely cancelled as above, but there will remain the term 



(4) 



representing the activity of the impressed forces, in addition to 

 the surface-integral. If we extend the integral in (3) to a space 

 to whose boundaries electric and magnetic actions do not extend, 

 since the integrand in the surface-integral vanishes we have 



(s) +*- 



as the equation of activity (cf. 64 (6)), whose H is the present E, 

 while H of (5) is the 2F of 64, (8).) 



If the fields are not zero at the surface S, the equation (3) 

 shows that the energy in the volume will be accounted for by 

 supposing that a quantity of energy 



per unit of surface S enters the volume r in unit time. We may 

 therefore call the vector 



