( 407 ) 



§ 5, We may now pass to a theorem which T liave formerly 

 proved in a somewhat more cumbrous and less general way. In 

 order to arrive at it, we have to use the complex vectors, supposing 

 at the same time the existence of magnetomotive forces; we have 

 therefore to apply the formulae (5) and (6). 



We shall consider two different states with the same frequency n, 

 both of which can exist in the sjstem of bodies. The symbols ^, J^, 

 etc. will be used for one state and the corresponding symbols, 

 distinguished by accents, for the other. We shall proceed in a way 

 much like the operations of the last paragraph, with this difference 

 however, that we shall now combine quantities relating to one state 

 with quantities belonging to the other. 



We shall start from the relation 



c \ (.fp'. rot t) - (^•. rot lp')| = - ( ly. 5?) - ((?. ^'). 

 Here the expression on the left is equal to 



and on the other side we may put 



( fV. ^) = in (.p'. 55) = in ( iq) ^'. 53) - ( p',. è), 

 {I. e') = ( (p) (£. S') - {^e. S'), 



SO that we find 



c div [^. S^^^ = - in ( iq) ^'. ^) - ( (/>)(£ . (è') + (.rp',. 55) + ((?,. (S ). 

 The theorem in question is a consequence of this formula and the 

 corresponding one that is got by interchanging the quantities belonging 

 to the two states; we have only to subtract one equation from the 

 other. Since, by (8) and (7) 



( {q) 5>'. ^) = ( (?) 5S. ^') and ( (p) (?. (è') =: ( (/>) (S'. (è), 

 we lind in this way 

 c [divit. Xp'] - div [(2'. .rp]| =: ( ry, ^^) _ ^ rp,. %') + (£•,. d') - {i\. d). 



We shall finally multiply this by an element of volume dS, and 

 take the integral of both sides over the space within a closed surface 

 0. If we denote by n the normal to the latter, drawn outward, the 

 result will be 



c(\ [^-. Sn^ - r^'. •C>]. I da =Jj (.ip'e/^)-(/Pe.^i>') f (ï-e.e') -{d'c.d)]dS (25) 



§ 6, There are a number of cases in which the first member of 

 this equation is zero. 



a. E. g. we may suppose the system to be limited on all sides 

 in such a way that it cannot exchange rays with surrounding bodies; 

 we can realize this by enclosing the sjstem '\\\ an envelop that is 



28* 



