280 Prof. A. Gray on the Steady Flow of 



Any tivo points in a linear system ivhich are at different 

 potentials may be joined by a wire without altering in any way 

 the state of the system, provided the wire contains an electromo- 

 tive force equal and opposite to the difference of potential between 

 the two points. For the wire before being joined will, in con- 

 sequence of the electromotive force, have the same difference 

 of potentials between its extremities as between the two 

 points ; and if the end of the wire which is at the lower po- 

 tential be joined to the point of lower potential, it will have 

 the potential of that point and no change will take place in 

 the system. The other end will then be at the potential of 

 the other point, and may be supposed coincident with that 

 point, without change in the state of the system. The new 

 system thus obtained plainly satisfies the principle of conti- 

 nuity and the other theorem assumed above, and is therefore 

 possible ; and it can be proved that it is the only possible 

 arrangement under the condition that the state of the original 

 system shall remain unaltered. 



As a particular case, any two points in a linear circuit 

 which are at the same potential may be connected either 

 directly or by a wire of any resistance, without altering the 

 state of the system. 



Further, it follows that if an electromotive force in one con- 

 ductor of a linear system produces no current in another conductor 

 of the system, either of the conductors may be removed without 

 altering the current in the other. For let one conductor be re- 

 moved : the potentials at the points of the system at which it 

 was attached will in general then be altered. And since any 

 two points in a linear system between which there is a differ- 

 ence of potentials may, without altering the state of the system 

 in any way, be joined by a wire which contains an electromo- 

 tive force equal and opposite to the difference of potentials, 

 we may suppose the conductor replaced with an electromotive 

 force in it equal to the difference of potential now existing 

 between the two points, and its presence or removal will not 

 affect the current in any part of the system. But the same 

 result may be attained, of course, without removing the con- 

 ductor, by simply placing within it the required electromotive 

 force ; and this by hypothesis does not affect the current in 

 the other conductor. Hence the removal of one conductor 

 does not affect the current in the other. 



If A, B, C, D be four points of meeting in a network of 

 linear conductors, in one wire of which joining AB there is 

 an electromotive force, while CD is connected by one or more 

 separate wires, the network can be reduced to a system of six 

 conductors arranged as in the figure, and such that the wires 

 AB ; CD, the currents in them, and the potentials at their ex- 



