338 ELECTROKINETICS. [PT. II. CH. VIII. 



flow which fill the portion A of the conductor continue in the 

 portion B, while the remainder leave them to traverse the portion 

 C. Then if we consider successive equipotential cross-sections 

 beginning in the portion A we shall finally reach an equipotential 

 which is divided into two parts, one lying in B and the other in C. 

 The last equipotential which does not break up into two consists 

 of two parts touching each other and touching the surface of the 

 fork of the conductor in a common point. This point, and this 

 equipotential surface may be taken to define the branching of the 

 conductors, and the surface will be taken for the common electrode 

 for the three portions A, B, and C. In a similar manner we may 

 have a conductor branching into any number of portions at a 

 common equipotential surface. Consider now any network of con- 

 ductors forming a figure of any degree of connectivity. The 

 distribution of current and potential is determined when the im- 

 pressed electromotive forces are given. If the equipotentials of 

 embranchment are given, we may consider each conductor r between 

 two successive surfaces of embranchment as a separate conductor, 

 to which we may apply Ohm's Law, 



(1) E r +V r -V r ' = B r I r , 



for 7 r , the total current in the branch, is perfectly defined. 



At every surface of embranchment p the equation of continuity 

 holds, so that if we call the currents in the s different branches 

 positive if they all flow away from the embranchment, 



(2) 7 pl + / p2 + I ps = 0. 



For every conductor there is an equation of the form (i), and 

 for every embranchment one of the form (2). The equations are 

 all linear in the currents in the different branches and the poten- 

 tials of the embranchments. They therefore suffice to determine 

 all the currents and potentials, in terms of the resistances and 

 impressed electromotive forces, except that the potentials may 

 contain an arbitrary constant. This is determined if the potential 

 at any one equipotential surface is given. 



In the above we have assumed the equipotentials of embranch- 

 ment given. It is easily seen however that these surfaces will 

 vary in form as the impressed electromotive forces vary. Suppose 

 for instance that an electromotive force be impressed in the branch 

 C of Fig. 68 so as to make the total current in that branch 



