whose electromotive force is insufficient to decompose water, becomes competent to do so by the intervention of a coil or electromagnet. Thus, let the primary wire of a Ruhmkorff coil be connected in the usual manner with the battery, and the electrodes of the voltameter (which may consist of a test-tube containing dilute sulphuric acid into which dip platinum wires) with the points where in the ordinary use of the instrument the contact is made and broken. There will thus be always a complete conducting circuit through the voltameter; but when the contact is made the voltameter will be shunted, and the poles of the battery joined by metal. Now when the shunt is open the battery is unable to send a steady current through the voltameter, because, as has been shown by Thomson, the mechanical value of the chemical action in the battery corresponding to the passage of any quantity of electricity is less than that required for the decomposition of the water in the voltameter. When, however, the shunt is closed, a current establishes itself gradually in the coil, where there is no permanent opposing electromotive force, and after the lapse of a fraction of a second reaches its full value as given by Ohm's law. If the contact be now broken, there is a momentary current through the voltameter, which causes bubbles of gas to appear on the electrodes, and which is often (but not, I think, well) called the extra current. Allowing the rheotome to act freely we get a steady evolution of gas.

To this electrical apparatus Montgolfier's hydraulic ram is closely analogous. The latter, it will be remembered, is a machine in which the power of a considerable quantity of water falling a small height is used to raise a portion of the water to a height twenty or thirty times as great. The body of water from the reservoir flows down a closed channel to the place of discharge, which can be suddenly closed with a valve. When this takes place, the moving mass by its momentum is able for a time to overcome a pressure many times greater than that to which it owes its own motion, and so to force a portion of itself to a considerable height through a suitably placed pipe. Just as the electromotive force of the battery is unable directly to overcome the opposing polarization in the voltameter, so of course the small pressure due to the fall cannot lift a valve pressed down by a greater. But when an independent passage is opened, the water (or electricity) begins to flow with a motion which continues to accelerate until the moving force is balanced by friction (resistance), and then remains steady. At the moment the discharge-valve is closed (or, in the electrical problem, the shuntcontact is broken), the water, by its inertia, tends to continue moving, and thus the pressure instantly rises to the value re

quired to overcome the weight of the great column of water. The second valve is accordingly opened, and a portion of the water is forced up. Now the electrical current, in virtue of self-induction, can no more be suddenly stopped than the current of water; and so in the above experiment the polarization of the voltameter is instantly overcome, and a quantity of electricity

passes.

If no second means of escape were provided for the water in the hydraulic ram, the pipe would in all probability be unable to withstand the shock, and in any case could only do so by yielding within the limits of its elasticity, so as gradually, though of course very quickly, to stop the flow of water. The bursting of the pipe may properly be compared to the passage of a spark at the place where a conductor carrying an electric current is opened. Just as the natural elasticity of the pipe or the compressibility of the air in a purposely connected air-vessel greatly diminishes the strain, so the electrical spark may be stopped by connecting the breaking-points with the plates of a condenser, as was done by Fizeau in the induction-coil. Contrary to what might at first sight have been expected, the fall of the primary current is thus rendered more sudden, and the power of the instrument for many purposes increased. Of course the spark is equally prevented when the breaking-points are connected by a short conducting circuit, as in our experiment by the voltameter. In fact the energy of the actual motion which exists the moment before contact is broken is in the one case transformed into that of the sound and heat of the spark, and in the other has its equivalent partly in the potential energy of the decomposed water, partly in the heat generated by the passage of the momentary current in the voltameter branch.

nue.

The experiment will be varied in an instructive manner if we replace the voltameter by a coil (with or without soft iron), according to the resistance and self-induction of the latter. In order to know the result, we must examine closely what takes place at the moment when contact is broken. The original current, on account of its self-induction or inertia, tends to contiAt the same time the inertia in the branch circuit tends to prevent the sudden rise of a current there. A force is thus produced at the breaking-points exactly analogous to the pressure between two bodies, which we will suppose inelastic, one of which impinges on the other at rest. The pressure or electrical tension continues to vary until the velocities or currents become equal. All this time the motion of each body or current is opposed by a force of the nature of friction proportional to the velocity or current. Whether this resistance will affect the common value of the currents (or velocities) at the moment

they become equal, will depend on its magnitude as compared with the other data of the problem.

There is for every conducting circuit a certain time-constant which determines the rapidity of the rise or fall of currents, and which is proportional to the self-induction and conductivity of the circuit. Thus, to use Maxwell's notation, if L and R be respectively the coefficient of self-induction and the resistance, the L time-constant is =T. If the current c exist at any moment R in the circuit and fall undisturbed by external electromotive

force, the value at any time t afterwards is given by a=c.e. Any action which takes place in a time much smaller than 7 will be sensibly unaffected by resistance.

We see, then, that we may neglect the effects of resistance during the time of equalization of the currents, provided that the operation is completed in a time much smaller than the timeconstants of either circuit. And this I shall suppose to be the case. The value of the common current or velocity at the moment the impact is over will of course be given by the condition that the momentum, electromagnetic or ordinary, is unchanged. If L and N be the coefficients of self-induction for the main and branch circuits respectively, x and X the original and required currents, the analytical expression of the above condition is

It is here supposed that there is no sensible mutual induction between the two circuits.

The spark is the result of the excess of the one current over the other, and lasts until its cause is removed. Its mechanical value is the difference between that of the original current in the main circuit and that of the initial current in the combined circuit, and is expressed by

Exactly the same expression holds good for the heat produced during the collision of the inelastic bodies, which is necessarily equal to the loss of ordinary actual energy, at least if the per

manent change of their molecular state may be neglected. From the value X the current gradually increases or diminishes to that determined according to Ohm's law, by the resistance of the combined circuit. It may be seen from the expression just found that the resistance of the branch may be varied without affecting the spark, provided always that it is not so great in relation to the self-induction as to make the time-constant comparable in magnitude with the duration of the spark. The spark depends only on the comparative self-induction of the branch circuit, being small when this is small, and when this is great approximating to its full value Lx2.

These results are easily illustrated experimentally. I have two coils of thick wire belonging to an electromagnet, which for convenience I will call A and B. Each consists of two wires of equal length, which are coiled together. These may be called A A, B, B. When A, A, are joined consecutively, so that the direction of the current is the same in the two wires, we have a circuit whose self-induction is four times that of either wire taken singly. But if, on the contrary, the current flows opposite ways in the two wires, the self-induction of the circuit becomes quite insensible.

The main circuit may be composed of the wire A, (A, remaining open) into which the current from a single Daniell cell is led, and which can be opened or closed at a mercury cup. One end of the branch circuit dips into the mercury while the other communicates with the wire whose entrance or withdrawal from cup closes or opens the main circuit. In this way the coils of the branch may be said to be thrown in at the break.

the

If the branch is open, we obtain at break the full spark, whose value is La2. If the wire B, be thrown in, the spark is still considerable, having approximately the value Læ2 for N=L. And if B, B2 are thrown in, so that the currents are parallel, the spark is still greater and is measured by La2 × 4. But if the currents are opposed, the spark disappears, because now N=0; so that the addition of the wire B2, whereby the resistance of the branch is doubled, diminishes the spark. It is true that to this last case our calculation is not properly applicable, inasmuch as the time-constant of the branch is so exceedingly small. But it is not difficult to see that in such a case (where the self-induction of the branch may be neglected) the tension at the breaking-points, or more accurately the difference of potential between them, cannot exceed that of the battery more than in the proportion of the resistances of the branch and main circuits, so that it could not here give rise to any sensible spark. Soft iron wires may be introduced into the coils in order to exalt the effects; but solid iron cores would

allow induced currents to circulate which might interfere with the result.

In this form of the experiment there was no sensible mutual induction between the coils A and B. Should there be such, the result may be considerably modified. For instance, let the wire A, be thrown at the break into the circuit of A, and the battery. This may happen in two ways. If the connexions are so made that the currents are parallel in A, A,, there will be no sensible spark; but if the directions of the currents are opposed, the spark appears equal to the full spark La2.

And this is in accordance with theory. The current X is given by the same condition as before, which leads to the equation

Lx+Mx=(L+2M+N) X,

M being the coefficient of mutual induction between the two circuits. The spark is therefore

Lx2— } (L+2M+N) X2=

x2 L-M
2 2

as N=L.

Now in the first-mentioned connexion M=L very nearly, and in the second M-L; so that the observed sparks are just what theory requires.

With regard to those electrical phenomena which depend on the mutual induction of two circuits, it may be remarked that it is not easy to find exact analogues in ordinary mechanics which are sufficiently familiar to be of much use as aids to conception. A rough idea of the reaction of neighbouring currents may be had from the consideration of the motion of a heavy bar to whose ends forces may be applied. If when the bar is at rest one end is suddenly pushed forwards in a transverse direction, the inertia of the material gives the centre of gravity in some degree the properties of a fulcrum, and so the other end begins to move backwards. This corresponds to the inverse wave induced by the rise of a current in a neighbouring wire. If the motion be supposed infinitely small, so that the body never turns through a sensible angle, the kinetic energy is proportional to § (a2 + k2) x2 + 1⁄2 (b2 +k2)y2+(ab−k2)xy,

where a and b are the distances of the driving-points (whose velocities are x and y) from the centre of gravity, k2 the radius of gyration about the latter point. This corresponds to the expression for the energy of the electromagnetic field due to two currents,

La2 + Mxy + Ny2;

and if we imagine the motion of the driving-points to be resisted by a frictional force proportional to the velocity, we get a very tolerable representation of the electrical conditions.