Electric Currents in Netivorks of Conductors. 239 



of trough and diagonal wire is intermediate between the greatest 

 and least when it is in position removed either 90° or 0° from 

 AB. 



By using a circular glass canal filled with sulphate-of-zinc 

 solution, and a zinc diagonal electrode and amalgamated -zinc 

 electrodes at A and B, a variable resistance may be constructed 

 capable of being varied over considerable ranges perfectly 

 gradually and with no imperfect contacts. 



§ 13. Having illustrated, by the foregoing examples, the 

 methods of calculating both the currents in and resistances of 

 networks of any complexity, we return for a moment to some 

 general considerations. 



Consider a function formed of the sum of each separate 

 resistance in a network multiplied by the square of the current 

 strength flowing through it. This expresses the heat gene- 

 rated per second in the whole network by that distribution of 

 current. This is called the Dissipation Function of the net- 

 work. It represents the rate at which energy is being trans- 

 formed into heat or rendered unavailable. 



Write down the dissipation function for the network in 

 fig. 1. Call it H. Then 



H = B# + loc —y + H#— r + Gy + Lz—y' + As + Ju—y' 

 + Ki^ 2 + (B + E)u 2 + (F + G)w 2 + Mu^ 2 . 



Now the cycle equation for the cycle or mesh y is, by Max- 

 well's rule, 



(C + I + L + J)?/ — Ix — Lz — Ju=0, 



which is the same as 



Cy — 1%— y — liz— y— Ju—y = 0. 



And this is at once seen to be identically the same as the first 

 partial differential of the dissipation function with respect to 

 the cyclic symbol y, or is the same as 



where B represents partial differentiation ; and by writing 

 down the other cycle equations for each cyclic symbol or 

 imaginary current, #, y, z, &c, we can show that these cur- 

 rent-equations are respectively 



xdH L BH idH 



S2 



