Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/376

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334
LINEAR ELECTRIC CURRENTS.
[280.

Let be the potentials at respectively, and let be the quantities of electricity which enter the system in unit of time at each of these points respectively. These are necessarily subject to the condition of 'continuity'


(4)

since electricity can neither be indefinitely accumulated nor produced within the system.

The condition of 'continuity' at any point is


(5)


Substituting the values of the currents in terms of equation (1), this becomes


(6)


The symbol does not occur in this equation. Let us therefore give it the value


(7)

that is, let be a quantity equal and opposite to the sum of all the conductivities of the conductors which meet in . We may then write the condition of continuity for the point ,


(8)


By substituting 1, 2, &c. n for p in this equation we shall obtain n equations of the same kind from which to determine the n potentials

Since, however, there is a necessary condition, (4), connecting the values of there will be only independent equations. These will be sufficient to determine the differences of the potentials of the points, but not to determine the absolute potential of any. This, however, is not required to calculate the currents in the system.

If we denote by the determinant


(9)


and by the minor of we find for the value of


(10)


In the same way the excess of the potential of any other point,