( 406 ) 



The equation (5) now becomes 



(? + (T-e = («) (E - i (^) e (20) 



or, if we define a new vector 25 by means of tlie equation 



e = i), (21) 



(?-f. e-, = («)(5 + n(/?)5:) (22) 



In the deduction of tlie equation of energy we have to understand 

 by ^, (Fg, Sp and X) the real vectors. For these ^ve have the formulae 

 (1), (2) and (21), and besides, since q, a and [3 are real, the relations 

 (4) and (22). 



From (1) and (2) we may draw immediately 



c |(/p . rot (f) - (X-. rot /p)| = _ (.f) . ^) _ CI- . CS), 

 the left-hand member of which is 



div S, 

 if we define the vector © by the equation 



S = c [.£• . /p], (23) 



i.e., if we understand by it the vector product of *5 and Jp, multiplied 

 by c. 



In the right-hand member we have in the first place 



(.,.») =i|(.rp.S,, 



as may be seen from (4), if (8) is taken into account, and further, 



in virtue of (7), (21) and (22), 



1 A 

 {d . g) = ( («) 6 . 6) + -n- ( (i?) 1> . 5^) - i^-e . (5). 



Our equation therefore takes the following form, in which the 

 meaning of the ditferent terms is at once apparent, 



{^, . 6) = ((«) 6.6) + l^^^m^- ^) + ^ar ^'^ • ^^ + "^'^ ®- 



The first member represents the work done by the electromotive 

 force per unit of volume and unit of time; in the second member 



u' .= ((«) (J. (?) (24) 



is the expression for the quantity of heat that is developed per unit 



of space and unit of time. Further, — (.0. 33) is the magnetic and 



-n( (/?) 5^, 5>) the electric energy, both reckoned per unit of volume. 



The vector S denotes the fiow of energy, so that the amount of 

 energy an element of volume dS loses by this flow is given t)y 

 div ^dS. 



