168 170] STEADY FLOW IN CONDUCTORS. 337 



that each is in contact only with the preceding and succeeding, 

 the system forms a single current tube, and the current is the 

 same through any cross-section. The conductors are then said to 

 be connected in series. If the surfaces of contact are equipoten- 

 tial we may apply Ohm's Law to each conductor. The potential 

 at the entering electrode of the rth conductor being V r , and at the 

 issuing electrode V r ', we have 



(2) 



Adding these equations we have 



(3) F 1 + # 12 + # 2 3 ..... .+E n _ l<n -V n '=E+V 1 -V n ' 



so that if R be the resistance of the system, 

 (4) R = R, + R 2 ...... + JBn, 



or the resistance of conductors in series is the sum of their 

 individual resistances. If the conductors are linear the conditions 

 at the ends are sure to be fulfilled. 



170. Networks of Conductors. Kirchhoff's Laws. 

 We have so far considered conductors filling a singly-connected 

 space. In order to treat a conductor filling a multiply-connected 

 space we have only to reduce it to a singly-connected region by 

 the insertion of cross-sections, and it is easily seen that if the 

 difference of potential on the two sides of a cross-section is given 

 the potential is determined. These cross-sections are most 

 naturally taken as the surfaces of impressed electromotive force. 



LEO" 



FIG. 68. 



Suppose now that a conductor has in a certain region a forked or 



embranched form, as in Fig. 68. Then a portion of the tubes of 



w. E. 22 



