The lamp was made to form one side of a Wheatstone's bridge, the other two sides being two resistances of 1 ohm each, and a variable resistance. In one diagonal was a condensor and telephones. The alternating apparatus, as in the last experiment, was placed in the other diagonal.

In this case the silence was not so good as in the former experiment, but there was a decided minimum when R was about 6 ohm.

The values of the back E.M.F. from these experiments with ordinary arcs agree closely with those found by Von Lang and Arons for small arcs. This constancy leads to the conclusion that the back E.M.F. is due to some process which goes on in the arc independently of the magnitude of the current which the arc is taking. Whether this process is the vaporisation of carbon and whether the back E.M.F. has its seat at the crater or in other parts of the arc, remains yet to be determined.

Note on Mr. Frith's paper "On the True Resistance and on the Back Electromotive Force of the Electric Arc." By Arthur Schuster, F.R.S.

(Received March 14th, 1895.)

The good agreement which has been obtained by different observers in measuring the so-called "back electromotive force" of the voltaic arc seems to shew that there must be some real meaning in the result, but it would be rash to take that agreement as a proof that the arc is the seat of something analogous to electrolytic polarisation, as the term would seem to imply. At any rate it is of importance to be clear as to what is really proved and what assumed in such experiments as those described in the previous paper.

Consider part of a circuit lying between two points, P and Q,and only containing metallic conductors and electrolytes or voltaic cells. The difference of potential e between the points P and Q, is connected with the current strength i, the resistance r, and the sum of electromotive forces included (E) by the equation

If we vary the current and observe the change in the fall of potential, a second equation is obtained which, together with (1), determines both E and r. Thus, if the changes are small, we obtain from (1)

This method of determining the electromotive force included in a circuit by means of measurements taken at the terminal points gives correct results as long as Ohm's law holds for all parts of the conducting system. But although we cannot speak of Ohm's law when the electric current passes through a gas, it is yet assumed in all discussions of this so-called back electromotive force that equation (2) gives what is called the "true" resistance. We need only analyse the various devices which have been adopted to separate resistance and electromotive force in the arc to see that they all depend on defining the resistance as the ratio of an increase of electromotive force to an increase of current. Generally the arc is maintained by an independent battery, and is increased or diminished by a small additional current; the change produced is then made to affect the measuring instruments. The difficulty is inherent, and I do not see how it can be overcome by any measurements made outside the arc. A special case will shew that the objection raised is a serious one. I will imagine that an experimenter wishes to decide whether there is a "back electromotive force " in a vacuum tube and in order to do so adopts such methods as have been assumed to give correct results for the voltaic arc. A series of measurements supplied by Hittorf* will allow us to calculate the results he would get.

In the following table the first column gives the number of the experiment, the second the measured difference of potential between the ends of a vacuum tube, and the third the corresponding current in micro-amperes.

Wied. Ann., Vol. XX. p. 727.

In the fourth and fifth columns I have placed the differences Ae and Ai of electromotive force and currents in successive experiments after the third, the fall of potential in the first three being practically constant. The next column gives r and E calculated by means of the equations

where for i the arithmetical mean of the currents observed in two successive experiments is taken.

The last column shews a remarkable consistency; the first six experiments have given practically identical values for the "back electromotive force," although the current has increased in the ratio of 1:34. The mean of the two last experiments also gives the same result, so that increasing the current nearly fifty times we always obtain the same value for E. Such a coincidence, our imaginary experimenter would argue, must prove the correctness of his method, and he would further draw the remarkable conclusion that the "true" resistance of his vacuum tube was zero up to a certain current intensity, and then suddenly rose and remained nearly equal to 4,200 ohms. Now

we know from investigations made inside the vacuum tube that all these conclusions would be incorrect, and the consideration of this case brings out the weak spot in the argument which has been applied to the voltaic arc. It is obviously impossible to separate a fall of potential due to an electromotive force from one due to other causes, as long as the fall is independent of the current intensity. Now we know that in vacuum tubes the fall of potential in the glow, as long as it does not cover the electrode, is constant, and also that the fall in the positive part of the discharge is, to a great extent, independent of the current. Hence, as long as the glow has room to expand on the kathode, the difference of potential between the electrodes does not rise, although the current may increase considerably. This was the case in the first three experiments quoted, and here we have the explanation why the whole difference of potential appears as "back electromotive force" and the "true" resistance seems to vanish.

Our familiarity with Ohm's law renders it difficult for us to imagine that the gradient of potential may remain the same though the current intensity varies, but experiments show that it is often the case in a gas. If we imagine the current to be due to a diffusion of ions it is easily seen that if more ions are set free at the electrodes, the same fall of potential must mean a greater current strength.

We should be cautious, therefore, in the interpretation of the experimental results obtained by Von Lang, Arons, and Mr. Frith.

Mr. Firth is correct in stating, at the end of his paper, that the constancy of the results obtained shews that it is due to some process which is independent of the current strength, but that process need only be a dissipation of energy, according to Joule's law, such as takes part in the positive part of the discharge in a vacuum tube, and does not necessarily imply any chemical or physical work done.