171, 172] STEADY FLOW IN CONDUCTORS. 841 



and Arg the minor of C rs , we have A a symmetrical determinant, 

 and A rs = A sr , since C pq G qp . The solutions of the equations (6) 

 are of the form 

 (7) A . V t = A lt (C U E U + C 12 # 12 + ...... + C ln E m ) 



+ A 2i (G 21 E 21 + 632^22 + ...... 4- @znEm) ...... 



4- An_ lf (C n -i, i J&ti-i, i + @n-i, 2 ^w-i, 2 ...... 



Inserting these values of the potentials in the equations (4), 

 we obtain the currents in all the branches as linear functions of 

 the impressed electromotive forces in the branches. Picking out 

 the terms containing E rs or its negative E sr in the current I pq we 

 obtain 



/ox Hpq 



In like manner the coefficient of E pq ^ in I rs is 



(9) 



But since A rs = A^, etc., this is equal to 



n 



Consequently the current produced in a branch pq as a result 

 of introducing an electromotive force E in a branch rs is the 

 same as the current produced in the branch rs on introducing an 

 equal electromotive force into the branch pq. This theorem is 

 analogous to the reciprocal property of electrified conductors given 

 in 136. If 

 ( 10) & pr + A 9S = & qr + A^ s , 



an electromotive force applied in one branch produces no current 

 in the other, and the conductors are said to be conjugate. 



172. Heat developed in the System. If we denote the 

 coefficient 



G pq C rs (A^ 4- A 9S - A 9r - A^ s ) by C pqrs> 



we have 



r=l g= 



