Self-induction of Wires. 69 



(5) Impressed Force in the Quadrilateral. General Property 

 of a Linear Network. — In my remarks on p. 190, vol. xxiii., 

 relating to the behaviour of batteries when put in the quadri- 

 lateral, I, for brevity in an already long section, left out any 

 reference to the theory. As is well known, in the usual 

 Christie arrangement (see figure, above) the steady current in 

 5, due to an impressed force in any one of 1, 2, 3, 4, is the 

 same whether 6 be open or closed, if a steady impressed force in 

 6 give no current in 5. But the distribution of current is not 

 the same in the two cases ; so that, when we change from one 

 to the other, the current in 5 changes temporarily; as may be 

 seen in making Mance's test of the resistance of a battery, or 

 by simply measuring the resistance of the battery in the same 

 way as if it had no E.M.F., using another battery in 6, but 

 taking the galvanometer zero differently. We, in either case, 

 have not to observe the absence of a deflection ; or, which is 

 similar, the absence of any change in the deflection ; but the 

 equivalence of two deflections, at different moments of time, 

 between which the deflection changes. Hence Mance's method 

 is not a true nul method, unless it be made one by having an 

 induction-balance as well as one of resistance ; in which case, 

 if the battery behave as a mere coil or resistance, which is 

 sometimes nearly true, especially if the battery be fresh, we 

 may employ the telephone instead of the galvanometer. 



The proof that the complete self-induction condition, 

 Z 1 Z 4 = Z 2 Z 3 , where the Z's stand for the generalized re- 

 sistances of the four sides of the quadrilateral, when satisfied, 

 makes the current in the bridge-wire due to impressed force 

 in, for example, side 1, the same whether branch b* be open or 

 closed, without any transient disturbance, is, formally, a mere 

 reproduction of the proof in the problem relating to steady 

 currents. Thus, suppose 



C 5 = ^\ (12<0 



where e l is a steady impressed force in side 1, and A and B the 

 proper functions of the resistances, in the case of the common 

 Christie, but without the special condition R 1 R 4 = B 2 B 3 , which 

 makes a resistance-balance. Then we know that, if we intro- 

 duce this condition into A and B, the resistance E 6 can be 

 altogether eliminated from the quotient A/B, making C 5 due 

 to e x independent of B 6 . 



Now, in the extended problem, in which it is still possible 

 to represent the equation of a branch by V = ZC, wherein Z 

 is no longer a resistance, we have merely to write Z for R in 

 the expansion of A/B to obtain the differential equation of G 5 ; 



