[ 489 ] 



XLIX. Note on an Elementary Treatment of Conducting 

 Networks. By L. R. "Wilberforce, M.A., Professor of 

 Physics at University College, Liverpool *. 



IT may he worth while to point out that the well known 

 reciprocal relations between the parts of a conducting 

 network can be readily established without an appeal to the 

 properties of determinants. 



Let A, B, 0, D, . . ., be a number of points connected by 

 conductors AB, AC, AD, . . .. BC, BD, . . ., CD, . . ., of resist- 

 ances Rab, . . •, and suppose that currents Qa, . . ., are led 

 into the network at the points A, . . ., from without, subject 

 to the condition Qa+Qb+ . .. = 0, and that internal electro- 

 motive forces, Eab, • • •? act in the conductors in the directions 

 AH,.... Let the currents in the conductors be Cab, • • •, 

 and let the potentials at A, . . ., be Va, .... The fact that 

 there is no continuous accumulation of electricity at any point 

 gives us a series of equations whose type is : — 



Qa = C A b + Cac + (1) 



and the application of Ohm's law gives us a series whose 

 type is 



RabCab-Va-Vb + Eab (2) 



Suppose now that a different system of external currents, 

 Q'a, • • ., and internal electromotive forces, E'ab, • • •? are 

 applied to the same network, and let the consequent currents 

 and potentials be denoted by accented letters. Equations 

 similar to (1) and (2) will, of course, hold for these quantities. 

 Multiplying each equation of series (2) by the correspond- 

 ing current in the second system and adding, we obtain : — 



XRab Cab C'ab = 2C'ab (Va-Vb) +2E A b Cab. 



Now, remembering that C'nm = — C'mn, the coefficient of 

 any potential Vm on the right-hand side of this equation is 

 easily seen to be C'ma + C'mb +■ . . ., and thus is Q'm. The 

 equation thus becomes : — 



2Rab Cab C'ab=SV a Q'a + XB A b Cab. 

 By a similar process we obtain 



2Rab Cab C' A b=2V'a Qa + 2E' A b Cab- 

 [If the accented system is made to coincide with the unac- 

 cented system, we obtain SRab C 2 ab = SVa Qa + 2Eab Cab, 

 the equation of activity.] 



* Communicated l>v the Physical Society: road Jan. 23, 1903. 



