170, 171] STEADY FLOW IN CONDUCTORS. 339 



zero. Then all the tubes of flow in A pass into B and the equi- 

 potential surface of embranchment is as it were sucked up as 



FIG. 68 a. 



shown in Fig. 68 a. The conductor B is now longer than before, 

 and we accordingly see that the resistance of a branch is not 

 constant, but depends upon the electromotive forces. This diffi- 

 culty immediately disappears if the conductors are linear, when 

 the surfaces of embranchment reduce to points, where the several 

 conductors join. The resistance is then between definite points, 

 and the above linear equations determine the distribution of 

 currents. 



Kirchhoff, who first treated the general problem of a network of 

 linear conductors*, eliminates the potentials by adding the equa- 

 tions of the first kind above for any group of conductors of the 

 series forming a closed circuit. The potentials thus disappear, 

 and for the circuit we have the equation 



(3) E, + E+.. ..+E n -- 



\j/ j > n 



This and the equations (2) for the junctions are generally 

 referred to as the equations of Kirchhoff s two Laws. Maxwellf 

 treats the problem in the following more symmetrical form. 



171. MaxwelPs treatment of Networks. Consider n 

 points of junction, each of which, in the most general case, is 

 connected with each of the others by a conductor. The number of 

 conductors in this case is n(n 1)/2. If some of the conductors 

 are lacking this will be expressed by putting the conductivities 



* Kirchhoff. "Ueber die Auflosung der Gleichungen, auf welche man bei 

 der Untersuchung der linearen Vertheilung galvanischer Strome gefiihrt wird." 

 Fogg. Ann., Bd. 72, 1847. Ges. Abh., p. 22. 



t Maxwell, Treatise, 280. 



222 



