158*.] Let a number of conductors ABZ, ACZ, ADZ be arranged side by side with their extremities in contact with the same two points A and Z. They are then said to be arranged in multiple arc. Let the resistances of these conductors be R1 , R2 , R3 respectively, and the currents C1 , C2 , C3 , and let the resistance of the multiple conductor be R, and the total current C. Then, since the potentials at A and Z are the same for all the conductors, they have the same difference, which we may call E.

We then have E = C1 R1 = C2 R2 = C3 R3 = CR, but C = C 1 + C 2 + C3 , whence 1 1 1 1

. R R1 R2 R3 (7) Or, the reciprocal of the resistance of a multiple conductor is the sum of the reciprocals of the component conductors.

If we call the reciprocal of the resistance of a conductor the conductivity of the conductor, then we may say that the conductivity of a multiple conductor is the sum of the conductivities of the component conductors. Current in any Branch of a Multiple Conductor.

From the equations of the preceding article, it appears that if C1 is the current in any branch of the multiple conductor, and R1 the resistance of that branch, R C1 = C , (8) R1 where C is the total current, and R is the resistance of the multiple conductor as previously determined.

Kirchhoff has stated the conditions of a linear system in the following manner, in which the consideration of the potential is avoided.

RESISTANCE OF A WIRE

1. Condition of ‘continuity.’

At any point of the system the sum of all the currents which flow towards that point is zero.

2. In any complete circuit formed by the conductors the sum of the electromotive forces taken round the circuit is equal to the sum of the products of the currents in each conductor multiplied by the resistance of that conductor.

Longitudinal Resistance of Conductors of Uniform Section.

159*. Let the resistance of a cube of a given material to a current parallel to one of its edges be ρ, the side of the cube being unit of length, ρ is called the ‘specific resistance of that material for unit of volume.’

Consider next a prismatic conductor of the same material whose length is l, and whose section is unity. This is equivalent to l cubes arranged in series. The resistance of the conductor is therefore lρ.

Finally, consider a conductor of length l and uniform section s. This is equivalent to s conductors similar to the last arranged in multiple arc. The resistance of this conductor is therefore R= lρ . s When we know the resistance of a uniform wire we can determine the specific resistance of the material of which it is made if we can measure its length and its section.

The sectional area of small wires is most accurately determined by calculation from the length, weight, and specific gravity of the specimen. The determination of the specific gravity is sometimes inconvenient, and in such cases the resistance of a wire of unit length and unit mass is used as the ‘specific resistance per unit of weight.’

If r is this resistance, l the length, and m the mass of a wire, then R= l2 r . m