Kirchhoff has enunciated in the form of two 'Laws' the principles that must guide us in such an investigation.

Law I.-If any number of conductors meet at a point, and if all currents flowing to the point be considered +, and ail currents flowing from the point be considered, and if the condition of things be steady, or the potential at the point be not altering, then the algebraic sum of the currents meeting at the point must be zero. Or

Σ. C = 0.

Law II.—Let us suppose there to be such a net-work of conductors as that imagined above; there being cells of various E.M.F.s, and turned various ways, in this net-work.

If we imagine ourselves to start from any point in this net-work and to make a circuit through the conductors back to our startingpoint, we shall have passed through conductors of various resistances, shall have passed through various cells whose E.M.F.s are directed so as to drive à current against or with us, and shall have found various currents, some with us and some against us.

As long as we are passing along a single conductor 1, to which there are no outlets, the current has some fixed value C1 ; but on passing a point where two or more conductors meet we may find a different current C2, which will remain constant over the next piece of resistance r1⁄2 up to the next place where two or more branches meet. We can thus divide our circuit into portions of resistances 71, 72, 73, &c., respectively; along each of which will be a current C1, C2, C3, &c., respectively. If we call these currents + or -, according as they flow with us or against us respectively; and if we call the E.M.F.s that we encounter + or -- according as they tend to drive a current with us or against us respectively; then it can be shown, by successive applications of Ohm's law to these different portions of our circuit, that in the complete circuit we must have.

where e,

Ciri + C2 r2 + C3 r3 + = e + e' + e" +

e', &c., are the various E.M.F.s that we pass ; attention being paid to signs both of the C.s and of the e. s.

These E.M.F.s e, e', e'', may occur in any way in the circuit, and do not necessarily (or indeed usually) occur in the pieces of circuit 1, 2, &c., respectively.

This law is usually expressed thus.

'In any complete circuit, . CR = Σ. e.'

Proof of Kirchhoff s second law.—We will here indicate the manner in which Kirchhoff's second law may be proved. The figure represents part of a net-work of conductors, in which are introduced any

D

F

11

e

e

number of battery-cells, e, e, eg, &C. Between any two consecutive junctions, as A and B, the current has some constant value CAB; and between B and the next junction C there will be a constant current CBC, which will in general be different from CAB.

The total resistance of conductor and cells between A and B is represented by rAB.

The symbols e1, e, &c., represent the numerical values of the various E.M.F.s,

and we will suppose the signs of these E.M.F.s to be also included in these symbols.

The potentials of the points A, B, C, &c., will be represented by VA, VB, &c. Then, by successive applications of Ohm's law, we have

These two laws may be applied in any complicated system of E.M.F.s and of conductors, in such a way as to determine the currents in all the branches, giving us always n independent equations to determine n unknowns.

As a simple illustration we may take the case of fig. iii. § 9, where there is but one E.M.F. E. Let us suppose that E, R, r1, r2, &c., are known, and that it is required to find C, C1, C,, &c.

Applying Law I. to the point A (or B) we have

C = C1 + C2 + C3 + C1 •

(i.)

Applying Law II. successively to the circuits RAr, BR, Ar, Br, A, Ar, Br, A, Ar, Br, A, we get the equations

4

These are five independent equations, and will enable us to determine the five unknown currents.

However, the method employed in § 9 was in this case simpler to

pursue.

§ 13. Maximum Current with a given Battery.-If we have a battery of n cells, each of E.M.F. E, and of internal resistance B, it is of great importance to arrange them so as to drive the largest possible current through a given external resistance r.

If n has factors m and 7, we may couple the cells so as to have m in series and parallel. We thus get a battery of E.M.F. m E, and of internal resistance

m B

==

The current

If we investigate either algebraically, or by means of the calculus, the arrangement that will give a maximum value of C, we find that this occurs when.

Or, we obtain the maximum current with a given battery when it is so arranged as to make the internal resistance as nearly as possible equal to the external resistance. (For proof, see note.)

It is therefore convenient to have a battery in which the number of cells is a number having several pairs of factors.

Example. We will here assume current, E. M. F., and resistance to be expressed in the units, ampères, volts, and ohms respectively; these are units which will be spoken of later.

Let us have n =
24, E

= 2 volts, B = 2 ohms, and r = Then we have the following arrangements possible.

3 ohms.

It is needless to continue, for it is clear that (iv.) gives us the greatest current. And case (iv.) is that in which the internal resistance of the battery Of course we cannot in general

is 3 ohms, or equals the external resistance.

so subdivide our battery as to make the equality exact.

In such extreme cases as those of § 6 we approximate most nearly when the internal resistance is made as great as possible in case (I.), and as small as possible in case (II.).

Note.-Froof of the above rule.-Let the number of cells be m, and let

Hence C is greatest when the denominator is least, since we cannot alter the value of the numerator by varying the arrangement.

This will be the case when the term (m B – Ir)2, which must be positive, m B disappears; and this again will be the case when m B = Ir, or when r = But is the external, and m B

is the internal, resistance; hence our theorem

is proved.

CHAPTER XIV.

MEASUREMENT OF RESISTANCES AND OF E.M.F.S.

§ 1. Preliminary, on the Units Employed.-In Chapter V. we explained the unit of quantity of electricity that is employed in electrostatics; and in Chapter X. we explained the unit employed in measuring differences of potential. Both these were based upon the absolute C.G.S. system of units, and started from the electrostatic repulsion between two very small charged spheres. We might build upon these two units a further system of derived units for current or quantity per second, E.M.F., resistance, &c. But such a system would only have a theoretical interest, and would not be the most convenient system to employ in dealing with currents and with the phenomena accompanying currents; for in considering these phenomena we are rarely concerned with electrostatic actions.

The most important class of phenomena, accompanying electric currents, are the magnetic actions of a current. Hence, in 'current electricity' we employ a system of units, based on the C.G.S. system, and starting from the action of a current on a magnetic pole.

This system will be fully explained in Chapter XVII. Here we will merely remark that the absolute units of E. M. F. and of R in this system are so small that they are inconvenient for practical purposes. We therefore use generally

(i.) As unit of E.M.F., the volt. This is almost exactly the E.M.F. of a cell consisting of a Cu and a Zn plate immersed in a solution of ZnSO4. This unit is 100,000,000 (or 108) times the absolute unit of E.M.F. In somewhat the same way we use a kilomètre, instead of the centimètre, in large measurements of length.

(ii.) As unit of resistance, the ohm. This is the resistance of