only the difference of potential between the ends of, but also the current in, that portion of the circuit.

It has been pointed out by Lord Rayleigh (Phil. Mag. May, 1886) that the resistance offered by a conductor to the passage of a current through it is greater the smaller the period of alternation. This variation is due to the fact that as the alternation increases in rapidity the current is more and more confined by inductive action to the outer strata of the conductor which is therefore virtually reduced in section. This is not to be confounded with the fictitious increase of resistance seen in the expression R2 + n2L2 (see p. 293 below) which arises directly from the electromotive force of self-induction, but is a real increase of the value of R for the current in question. A very useful table of the resistances of conductors at different periods of alternation drawn up by Mr. W. M. Mordey and given in his paper on Alternate Current Working (loc. cit. p. 282) will be found in the Appendix. The theoretical data from which the table has been calculated were given by Sir William Thomson as an addendum to his presidential address (1889) to the Institution of Electrical Engineers. An abstract of it is printed with the table.

Denoting by Am the mean value of the electrical activity in this part of the circuit, still supposing the self-induction of this part to be negligible, we have plainly

In the same way, since the value of the electrical activity at any instant is CR, we have from the results of experiments made by an electro-dynamometer,

that is, the mean value of the electrical activity is equal to the product of the square root of the mean square of the difference of potential, by the square root of the mean square of the current strength. It can therefore be determined by means of an electrometer and an electrodynamometer of negligible self-induction without its being necessary to know the resistance.

We shall now consider the case in which the selfinduction cannot be neglected. Let R be the total resistance in the circuit, C the current flowing in it at the time t, E the total electromotive force of the machine, and L the coefficient of self-induction, or the "inductance" (as, following a usage introduced by Mr. Cliver Heaviside, we shall henceforth call it) for the whole circuit, that is, the number which multiplied into dC/dt gives the electromotive force opposing the increase or diminution of the current. We shall suppose L a constant, although there can be no doubt that in some alternating machines its value is different in different positions of the armature. The iron cores of the field magnets act to a greater or less extent as cores for the armature coils, and as the magnetic susceptibility of iron is a function of the strength of the magnetizing current, Z, which is the magnetic induction through the armature produced per unit of its own current, must vary accordingly.

Still for certain alternators which have no iron in their armatures the variation of L with the position of the armature is slight.* It will also be assumed that there

* See the discussion on Mr. W. M. Mordey's paper on "Alternating Current Working" Inst. of Elect. Eng. May, 1889 (The Electrician, May 24, 31, June 7, Aug. 2, 1889).

are no masses of metal in which local currents can be On these assumptions the

generated moving in the field.

But by the law which we have assumed for the machine,

where e is a constant such that E。 is the maximum value of E for the given speed. Substituting in (24) we get

The term A e-tis only important immediately after the circuit is closed, and will therefore be neglected.

We may remark that if Z were equal to zero (27) would reduce to C = Eo/R. sin nt, which corresponds to (12) above.

From (27) we get for the mean current

Also for the mean square of the current strength as

given directly by an electro-dynamometer we have by (27)

From (27) we see that the effect of self-induction is to diminish every value of the current in the ratio of Eo/(R2 + n2 L2) to E/R, and to produce a retardation of phase which measured in time is /n seconds; that is, the resistance is virtually increased in the ratio (R2+n2L2)1/R, and the current in following the law of sines passes through any value en seconds after it would have passed through the corresponding value if there had been no self-induction. It is plain also that, for any finite resistance R, by diminishing T, that is, by increasing the speed of the machine, the current can, by (25), be made to approach the limiting value

e

C =
L

sin

(nt - 1) .... (31)

2

which is independent of the resistance, and has a retardation of phase of T/4 seconds, a quarter period of a complete alternation. Hence integrating over a half period from zero current to zero current again, and dividing by T/2 we get for the maximum mean current

To find the mean value Am of the total electrical activity in the circuit, we have by (25), and (27)

that is, the true mean value of the total electrical activity is equal to the mean square of the current strength multiplied by the total resistance in circuit.

It

It may be shown, from (33), by the method used in p. 85 above that the total activity in the circuit is greatest when R=nL, that is, for a given speed and a given value of L, the activity is a maximum when R=nL. must be observed however that for a given resistance R the activity is greater the smaller the value of T, that is, the greater the speed. When R has the value nL we have, by (27) € π/4; that is, the retardation of phase is then one-eighth of the whole period.*

€ =

If in the circuit there be two sources of electromotive force of the same period 7 but of different phases; for example, two machines driven so as to have the same period of alternation, the solution here given applies. For the two electromotive forces combine to give a single electromotive force of the same period as the components

*The conclusions as to maximum work and retardation of phase, as well as most of the theoretical results stated above as to the action of alternating machines, were first we believe given by M. Joubert, Comptes Rendus, 1880. The problems of alternating machines joined in series, or in parallel, or as motors, were considered by Dr. J. Hopkinson in a lecture to the Inst. of Civil Engs. 1883, and in a paper "On the Theory of Alternating Currents," Soc. Tel. Engs. and Els., Nov. 1884. The principal results of this latter paper are reproduced below. See also Mr. Mordey's paper loc. cit.