1891.] Application to Periodic Electric Currents. 207 



and that through b + d is 



(q + c) -<^i /gx 



a+b+c+d 



The difference of potentials at the terminals of e, supposed to be 

 interrupted, is thus 



a+b+c+d 



be ad /HV 



or B = TT -7 ................. (') 



a+b+c+d 



By (4), (4'), (7) the relationship of ^i, i^ to T^I, ^2 is completely 

 determined. 



The problem of the bridge requires the determination of the cur- 

 rent yr 2 , as proportional to i r 1 , when 1r t = 0, that is, when no elec- 

 tromotive force acts in the bridge itself, and the solution is given at 

 once by simple introduction into (2) of the values A, C, B from (4), 



(4'), (7). 



If there be an approximate balance, the expression simplifies. 

 For be ad is then small, and B 2 may be neglected relatively to AC 

 in the denominator of (2). Thus, as a sufficient approximation in 

 this case, we have 



ad be 

 . 1^ a + b + c + d ,x 



" 



a+b+c+d J 



in agreement with the equation used by Mr. Heaviside for simple 

 resistances. 



The following interpretation of the process leads very simply to 

 the approximate form (8), and may be acceptable to readers less 

 familiar with the general method. Let us first inquire what E.M.F. 

 is necessary in the telephone branch to stop the current through it. 

 If such a force acts, the conditions are, externally, the same as if the 

 branch were open, and the current y^ in the battery branch due to an 

 E.M.F. equal to ^ in that branch is i^/A, where A is written for 

 brevity as representing the right-hand member of (4). The difference 

 of potential at the terminals of e, still supposed to be open, is found 

 once when y^ is known. It is equal to 



: 



ex (5) 



ere B is defined by (7). In terms of ^i the difference of poten- 

 tials is thus B^i/A. If e be now closed, the same fraction expresses 

 e E.M.F. necessary in e in order to prevent the generation of a 

 rrent in that branch. 



