336 ELECTROKINETICS. [PT. II. CH. VIII. 



the slope of the broken lines is the same, that the more evenly the 

 discontinuities are distributed the less is the maximum difference 

 of potential between various parts of the circuit. By making 

 the discontinuities small enough, we may therefore, without chang- 

 ing the current, make the differences of potential in the circuit as 

 small as we please. In the limit the electromotive forces would 

 be continuously distributed, and there would be no difference of 

 potential. In that case there would be no electrostatical electro- 

 motive force. Such a continuous distribution of electromotive 

 force may be produced by electromagnetic induction, the theory of 

 which will be given at length in Chapter XII. The existence of 

 a current does not, therefore, imply differences of potential. 



169. Conductors in Parallel and Series. By the defini- 

 tion of conductivity of a current tube, it is evident that the 

 conductivity of any number of current tubes between the same 

 two equipotential surfaces is the sum of their individual conduc- 

 tivities. Now those portions of the surface of any conductor 

 which are in contact with an insulator are portions of the sides of 

 a current tube, for there is no flux across them. If then two equi- 

 potential surfaces are given in such a conductor so that the 

 current flows in at one and out at the other, these surfaces are 

 known as electrodes for the conductor, and if the electrode sur- 

 faces of several conductors are brought into contact and kept 

 equipotential, the conductivity of the system is the sum of the 

 individual conductivities. The essential in this proposition is that 

 the contact of the several conductors shall not change the form of 

 the equipotential surfaces that have been called electrodes. This 

 condition is sure to be fulfilled if the conductors are linear, that 

 is if each conductor forms a tube of flow whose cross-section is 

 small enough to be neglected in comparison with its length. The 

 electrodes in this case reduce to surfaces of infinitesimal area, and 

 may be regarded as points. Conductors having two common 

 electrodes are said to be connected in parallel, or in multiple arc, 

 and for such the resistance R of the system is given by the 

 equation, 



(I) R^R^R, + ^' 



The resistance of the system is evidently less than any of the 

 separate resistances. If several conductors be placed in order so 



