( ^1^ ) 



Let s and s' be two elements of volume, caibitiarily chosen, h 

 one of the principal directions of the first element, Ji one of the 

 principal directions of the other, ah and «'/»' the coefficients relating 

 to these directions. 



In virtue of the electromotive force ^eh acting in s in the direction 

 h, there will be in s' in the direction k' a current (?/,' witli a certain 

 amplitude {jik^ ; by (31) the development of heat corresponding to 

 this current will be per unit of time 



\aid^h')\^ (43) 



Similarly, we may write 



-cthi^'i)'^ (44) 



for the heat developed in s on account of the current (i'/j produced 

 in this element in the direction li by the electromotive force acting 

 in s' in the direction h. 



Since each of the tlu-ee eleeti'omotive forces in s calls forth a 

 current in the element s' in each of its principal directions, there 

 will be in all nine expressions of the form (43). These must be 

 added to each other, as may be seen by observing that the total 

 development of heat, represented by (31), is the sum of three parts, 

 each belonging to one of the components of the current and tliat 

 the three electromotive forces in s are mutually independent. The 

 sum of the nine quantities will be the total amount of heat s' receives 

 from s, and in the same way we must take together nine quantities 

 of the form (44), if we wish to determine the amount of heat 

 transferred from s' to s. We shall have proved the equality of 

 tiie mutual radiations between the two elements, if we can show 

 that for any two principal directions, the expressions (43) and (44) 

 have the same value. 



Let us call tik and a'w the amplitudes of the electromotive forces 

 originating the currents whose thermal effects have been represented 

 by (43) and (44). Then, in accordance with (40) and (41), 



4 jr (• I /2 k «/j dn 4 :rr c j /2 k a'h' dn 



au' = / . . (45) 



s 11 y s 



Now, by the general theorem of § 7, a the amplitudes ((?a') and 

 (IÏ'//) ill (43) and (44) are proportional to rt/,s and a'/j'S'. Taking 

 into account the formula (45), we infer from this 



((i/,0^ : ((i'/,r = «a' S^ : a'/,' ' s" = a/, S : «'a' s', 

 an equation, which leads directly to the equality of (43) and (44). 



