therefore, substituting these values for zs', z T, s z and Us in the expression given above, we see that the increase in the power given to the external circuit by an increase in the current of q q' equals

qq'(V —b × qq' —b × oq),

but qq' is small compared with oq, since by hypothesis only a small change was made in the current, therefore the increase in the power given to the external circuit, produced by making a small increase in the current, equals approximately

QQ'(V-bxoq).

This expression will be positive as long as V exceeds bxo q, and will become negative when V becomes less than b xoqQ. But b xoq represents the P.D. used in sending the current through the cell; hence making a number of changes in the circuit so as to produce small increments in the current will increase the power given to the external circuit as long as V, the P.D. between the terminals of the generator, exceeds the P.D. used in the generator itself; but when these two become equal any further increase in the current will make the increase in the power delivered negative-that is, will begin to diminish the power given to the external circuit. Consequently, the external circuit that receives maximum power from a current generator, of fixed E.M.F. and resistance, is the circuit that makes the P.D. between the terminals of the generator equal to half its E.M.F.

This is the form of the law that it is most easy to test experimentally, and the apparatus shown in Fig. 169, page 366, may conveniently be used for this purpose.

The product of the readings of the ammeter A and the voltmeter v gives the power furnished to the external circuit, and if x, the resistance of this circuit, be altered and a set of simultaneous readings of the ammeter and voltmeter be taken, it will be found that the product of these readings will be a maximum when the deflection of the voltmeter is half the deflection that is obtained on breaking the external circuit—that is, when the P.D. between the battery terminals is half its E.M.F.

To test the law in its two previous forms would require the battery to be short-circuited, but the E.M.F. of even a so-called "constant battery" would change somewhat if the battery were short-circuited; while in the case of many useful batteries, although the E.M.F. is fairly constant when the resistance of the external circuit is two, three, or more times as great as that of the battery itself, it would fall considerably on the battery being short-circuited, and therefore would render an experimental test of the law in its first two forms very difficult to carry out.

It is to be observed that the preceding results are all generally true, whatever be the nature of that portion of the external circuit which we desire shall receive maxi

mum power. For example, the reasoning would be exactly the same whether the portion of the external circuit under consideration were composed of a variable resistance or whether it contained in addition a forward E.M.F. produced by some current generator that could be altered, or a back E.M.F. produced by some electrolytic cell, or by a running electromotor, the E.M.F. of which could be adjusted to bring the current to the required value.

From what precedes, then, we may conclude:

(1) If an external circuit be a simple resistance of x ohms, then in order that it may receive maximum power from a generator having a fixed E.M.F. of E volts and a fixed resistance of b ohms

x must equal b.

(2) If the external circuit contain in addition a forward E.M.F. of E' volts,

(3) If it contain instead a back E.M. F. of E' volts,

125. Arrangement of Part of the External Circuit to Receive Maximum Power.--If we desire that a current generator of fixed E. M.F. and resistance shall give maximum power to a portion of an external circuit-for example, if the generator be connected by long leads of fixed resistance lohms to a motor or to lamps at a distance, and we desire to arrange the motor or the lamps so that they shall receive the maximum power-then the fixed resistance of the leads must be added to the fixed resistance of the generator; hence for b in what precedes we must substitute b + l.

The three equations given at the end of §124 state the conditions under which an external circuit of resistance x ohms shall receive maximum power, this power being wholly employed in heating the circuit if it be a simple resistance, and partly so employed if the external circuit has any resistance at all. If, however, there be an apparatus in the external circuit which has a back E.M. F. of E' volts, and which, therefore, for example, produces a transformation of electric energy into mechanical or chemical energy, and if we desire to arrange this back E.M.F. so that this transformation of energy may be effected as rapidly as possible, and not merely that the apparatus shall receive the maximum power, then the solution is not the one previously given.

For if m be the resistance of this apparatus in ohms, b and being the resistances of the generator and the

N*

leads in ohms, then what we now desire is not that

A

{ E - - A (b + 1)} shall be a maximum, but that

and, by comparing this with the expression A (E = Ab), which was shown in § 124 page 388, to be a maximum

E

when A equalled we may conclude that the expression

26'

just found for A E' will be a maximum when

E

E= 2

Under those circumstances the expression for the rate of transformation of electric energy into non-heat energy in the apparatus becomes

which is the power expended by the generator in heating the circuit; so that here again the power utilised is half the total electric power which the generator develops.

Consequently, while it follows from the equation given at the end of § 124, page 239, that the apparatus in question will receive maximum power when

mechanical or chemical energy, on the contrary, will be effected most rapidly in the apparatus when

E

E' = 2

All the preceding conclusions are based on the assumption that the E.M.F. and resistance of the primary current generator are fixed and independent of the current flowing, and that the arrangement of the external circuit is the only variable under consideration. If, however, either the E.M.F. or the resistance of the primary generator change with alterations in the current, then the preceding conclusions will not be generally true, and the particular arrangement of the external circuit that will receive maximum power from the generator will depend on the exact way in which the E.M.F. and resistance of the generator varies with the current in each particular case.

Another class of problem also sometimes arises in practice-viz. one in which the conditions of the external circuit are fixed and cannot be altered, and it is the arrangement of the generator which gives maximum power to this external circuit that we desire to find out. In such a case it is clear that, since the conditions of the external circuit are fixed by hypothesis, the power given to it will depend on the current passing through it; hence the problem reduces itself to finding the arrangement of the generator that will send the greatest current through a given external circuit. To solve this problem some condition must necessarily be given limiting the power of the generator, since without such a condition all we can say is that the larger the E.M.F. and the smaller the resistance of the generator the greater will be the current it will send through the external circuit. When this condition is that the generator consists of a fixed number of galvanic cells of a given type the problem of finding the arrangement of these cells to produce the maximum current through the fixed external circuit will be found solved in § 163, page 548. Other