492 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. 



For every point of embranchment a we have an equation 



(2) / 



...... a ~, 



at 



the currents being now marked with double suffixes to denote the 

 points between which they run, as in 171, and e a denoting the 

 charge of the accumulator connected with the point, or zero if 

 there is no accumulator. These p equations are not all indepen- 

 dent, for adding them all together, every current appears in both 

 directions, so that the left-hand side in the sum is identically zero, 

 giving 



which is merely the statement that the total charge of the system 

 is unaffected by the flow of currents. There are accordingly p I 

 independent equations (2). 



For every accumulator K a we have an equation, 138 (10), 

 (4) V* =Pia,ei + p M ei + ...... + p ka e k = -5 . 



v@a 



From the equations (i) the F's may be eliminated by Kirch- 

 hoff's principle, 179. If, traversing any closed circuit, we add 

 the equations (1) for each wire, every V appears with both signs, 

 so that on the right we obtain the sum of the E's around the 

 circuit. We shall thus obtain as many equations as there are 

 independent meshes in the net, I p + 1. Other equations may 

 be obtained in the same manner by traversing any unclosed circuit 

 ending on two accumulators. All the potentials at embranch- 

 ments passed over are eliminated except those of the two ends. 

 The number of equations to be obtained in this manner is one less 

 than the number of accumulators, or k 1. We thus obtain in all 

 I p + k = n equations, and there are the same number of inde- 

 pendent variables. We may take as parameters to characterize 

 the system a set of currents, one circulating in each mesh, so that 

 the actual current in any wire is the sum or difference of the 

 currents in the two meshes to which that wire is common. The 

 time-integral of any mesh-current shall be taken for one of the 

 parameters q. Besides the lp + \ q's thus defined, we will 

 choose k 1 others, denoting the integral currents along any series 

 of wires joining the accumulators two and two, the whole series 

 forming a chain with two ends. The charge of any accumulator is 



