will not affect the circuit and the activity shown by the instrument may be taken as that existing when it is not applied.

But it is important to notice that the second condition may be fulfilled without an accurate measurement of power by the wattmeter. The currents through the two coils will have the same phase, but this may not be the phase of the difference of potential between the terminals if between these there be sensible self-induction in the main circuit; and the wattmeter would give too high as result. It is essential for accuracy that the current through the fine wire coil-system of the wattmeter should have the phase of the difference of potential, while that in the thick wire coil should have the phase of the main current.

The general problem of finding the ratio of the apparent activity as shown by the wattmeter to the true activity can be solved with great ease by aid of the theory given above. For let A, B be the points at which the terminals of the fine wire coil-system are attached to the main circuit; let R1, R2, L1, L2 be the resistances and inductances of the fine wire and thick wire circuits between A, B, and C1, C2 the currents in them, then by (27) if the difference of potential between the terminals A, B is E sin nt.

The current through the fine wire is therefore the same as if the resistance in its circuit between the points A, B

were without inductance, and the difference of potential E between A, B had the value

Eo R1(R12 + n2 L12). sin (nt - €).

Hence if A'm be the apparent activity in the main circuit between A, B

T

I

=

ERI

† (R ̧2+n2L{})} ({R }2+n2L®)} [ "sin(nt – e,)sin(nt – e,)dt

= n L1/(R12 + n2 L12), cos €1

with similar values for sin 2, COS €2.

[by (38)]

(60)

But if Am be the true mean activity then in the same

where T1, T2 are written for L1/R1, L2/R2 respectively, the socalled "time constants" of the two parts of the circuit.

Now in general T1 > T2, hence as a rule the wattmeter will give too high a result. If the time constants and the period of alternation be known, then Am can be calculated from the apparent activity by this equation.*

In any case in which a wattmeter is inapplicable, if the actual resistance of the portion of the circuit considered is known, and the mean square of the current can be measured with accuracy, the product of the two will, as shown above (p. 285), be the true mean value of the activity. This of course will be given in watts, if the resistance is taken in ohms and the current in amperes.

As we have seen above, the proper mean value of the current, and of the difference of potential, and therefore also of the activity, can be found for any part of a circuit in the case of negligible self-induction, either by means of an electro-dynamometer, or by means of an electrometer, when the resistance of the part of the circuit is known. When the resistance is unknown or uncertain, as for example in the case of incandescence lamps, the current and difference of potential may be measured for the lamp circuit in the following manner. A coil of german silver wire, having a resistance considerably greater than that of the lamps as arranged, constructed as described above (p. 188), so as to have no self-induction, is connected in series with a current-meter between the terminals of the machine so as to be a shunt on the lamps. The lamps are brought to their normal brilliancy, and the mean square C2 of the current through the german silver wire measured. If Rbe the resistance of this wire, including,

This result was given without proof by Prof. Ayrton in his remarks on "Testing the Efficiency of Transformers," Proc. Soc. Tel. Eng. and El. Feb. 1888.

For methods of determining time-constants, see the author's Theory and Practice, &c., vol. ii.

if appreciable, the resistances of the current-meter and its connections, and R be great in comparison with the coefficient of self-induction of the current-meter divided by T, we have for the mean square V2 of the difference of potential between the terminals of the lamp system, the value C2 R2. The current-meter is now employed to measure the whole current flowing to the lamps while their brilliancy is kept the same. Denoting the mean square of this current by C2, we have for the value Am of the mean activity spent in the lamp system, the equation

An electrometer may be used in the following manner to give the mean square of the current, and of the difference of potential for any part of a circuit whether containing motors, or arc lamps, or any arrangement with or without counter-electromotive force or self-induction. A coil of thick german silver wire (or to prevent sensible heating a set of two or more equal coils arranged in multiple arc) having no self-induction is included in the part of the circuit considered, so that the current to be measured also flows through the wire. The mean square of the difference of potential between the ends of this resistance is measured as described above (p. 279) by connecting one pair of quadrants of the electrometer to one end, and the needle and the other pair of quadrants to the other end, and the mean square C'2 of the current found by dividing by the square of the resistance of the wire. The mean square of the difference of potential between the terminals of the part of the circuit considered, is then found in the same manner. The product is not generally to be taken as the mean square of the activity in the part of the circuit

considered, for it is evident that in this case what is obtained is the value of

where V and C are the difference of potential and the current at any instant. The square root of this quantity is not generally the same thing as

the true mean value of the activity. This is, however, given directly by the following method.*

Let the two ends of the resistance coil of zero selfinduction and known resistance R be called A and B, and let the extremities of the portion of the circuit for which the measurements are to be made be called C and D. One of the pairs of quadrants is connected to A, the other pair to B, and the needle to C, and the reading d say taken. The quadrants remaining as they were, the needle is connected to D, and the reading ď taken. Now if at any instant V1 be the potential of A, V1⁄2 of B, V1 of C, and V1⁄2 of D, we get by (18) above

* This method is described by A. Potier, Journal de Physique, t. ix. p. 227, 1881, but was independently invented also by Prof. W. E. Ayrton, and Prof. G. F. Fitzgerald (see Prof. Ayrton on "Testing the Power and Efficiency of Transformers," Proc. Soc. Tel. Engs. and Els., Feb. 1888).