The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 13 of 35

Along the Gray's Inn-road, London, the English Post-office service placed a line of iron pipes buried underground carrying many telegraph wires. The United Telephone Company placed a line of open wires along the same route over the housetops, situated 80ft. from the underground wires. Considerable dis- turbances were experienced on the telephone circuit, and even Morse signals were read, which were said to be caused by the Continuous and parallel telegraph circuits. A very careful series of experiments,* extending over some period, proved unmis- takably that this was so, and that the well-known pattering disturbances due to induction are experienced at a much greater distance than was anticipated.

Experiments conducted on the Newcastle Town Moor extended the area of the disturbance to a distance of 3,000ft., while effects were detected on parallel lines of telegraph between Durham and Darlington at a distance of 10J miles. But the greatest distance experimented upon was between the east and west coast of the Border, when two lines of wire 40 miles apart were affected the one by the other, sounds produced at Newcastle on the Jedburgh line being distinctly heard at Gretna on a parallel line, though no wires connected the two places.

Distinct conversation has been held by telephone through the air without any wire through a distance of one quarter of a mile, and this distance can probably be much exceeded.

Effects are not confined to the air, for submarine cables half a mile apart in the sea disturb each other. It may well be doubted whether the inductive effects above described as taking place over very large distances above mentioned are not complicated by current leakage, but it has been abundantly established that inductive effects can be produced and detected between circuits separated by great distances.

Practical application of current induction across large air spaces has been made in the methods of carrying on tele- graphic communication with railway trains when in motion. There are two methods by which this has been accomplished. (1) The magneto-induction method, which was devised by

• Mr. W. H. Preece on "Induction between Wires and Wires" (The Electrician, Vol. XVII., 1836, p. 410 ; British Association Report, Birming- ham, 1886).

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Mr. L. J. Phelps and was tried about the year 1885 on a line about 15 miles long between Haarlem Eiver and New Eochelle Junction, in the United States. In the other system (2) the principle involved is that of electrostatic induction, and, after having been suggested in a more or less imperfect form by Mr. W. Wiley Smith in 1881 (U.S. Patent No. 247,127), has been worked out in great detail by Messrs. Edison and Gilliland.

In the magneto-induction system a telegraphic car attached to the train carries a great circuit of wire wound on a frame extending the whole length of the car, and so placed that one side of the windings is as near the track as possible and one side as high above as the height of the car will permit. Between the rails is laid down a fixed insulator conductor, and the fluctuations of a current in this last induce currents in the lower side of the large coil carried on the car. The secondary current so induced is detected by a telephone and by suitable interruptions. A Morse code of audible signals can be transmitted from the fixed conductor to the moving train. The signals are thus made to jump over the air space, and continuous communications can be kept up between a station or stations in connection with the fixed conductor and a person in the moving telegraph car.

Mr. Phelps used a conductor of No. 12 (A.G.) insulated wire, which was placed in a kind of small wooden trough mounted on blocks attached to the sleepers. The car con- taining the telegraphing instruments carried beneath its floors, about Tin. above the rail level, a 2in. iron pipe, in which was a rubbe: tube holding about 90 convolutions of No. 14 (A.G.) copper wire, so connected as to form a continuous circuit about a mile and a-half long, and presenting something like three-quarters of a mile of wire parallel to the primary line wire mounted between the metals. The instrument, consisted of a delicate polarised relay as a receiving instru- ment, which acted as a sounder, and a " buzzer," or rapid current-breaker, for transmitting signals by means of the Morse key, which were received at the station in a telephone. This arrangement was so far a practical success that Mr. Phelps was encouraged to proceed ; but meantime it was • discovered that the patent above referred to had already been

218 MUTUAL AND SELF INDUCTION.

issued, while Edison and Gilliland had also been working on similar lines. In Wiley Smith's specification no mention is made of a " buzzer," which turns out to be an important feature in the invention ; but the practical success of the experiments made is due to a combination of the devices of Phelps, Edison, and Gilliland. The latest system is an improvement on that of Phelps, briefly described above, in that it dispenses with the insulated line wire laid between the metals, and uses ordinary telegraph wire strung on what are known as short poles alongside the permanent way. The line wire is, in fact, stretched on poles about 16ft. high and at an average distance of 8ft. from the rails. On the Lehigh Valley Railroad, U.S., from Perth Junction to Easton, interesting experiments of this kind were made in 1887. As a rule the roof of the car, usually sheathed with metal, is available for securing the necessary electrical condition, but, where a metal covering is absent, all that is necessary is to attach a wire or rod to the roof and another to some portion of the metallic or rolling part of the coach in order to obtain " earth." The instruments are inserted in this circuit, and comprise a 12-cell chromic acid battery (the cells being 2in. wide by 4in. deep), which is closed on an induction coil having a primary of about 3 ohms and a secondary of about 500 ohms, and provided with an ordinary vibrating make-and- break. The messages are sent by means of a Morse key placed in the secondary circuit, this key being of the double- pointed kind with extra contact. The receiving telephone has a resistance of about 1,000 ohms ; but Mr. Phelps states that, even when wound so as to have a resistance of 10,000 ohms, the sound is quite clear, so high is the electromotive force of the induction on the roof. The car-roofs are frequently of metal — usually painted tin plates, sheet zinc, or galvanised iron, and these answer admirably as inductive receivers ; but where the roofs are of wood, covered with painted canvas, an iron or brass rod or tube, |in. in diameter, is carried along under the eaves throughout the length of the train. The metallic roof or the rod is connected by a wire to the secondary of an induction coil, while the primary of the coil is connected to the front contact of the double-pointed key, and through that with the battery. Opposite the core of the coil is the

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" buzzer," which transmits a series of impulses to the lin& whenever the key is worked. The extra contact, which is placed on the upper surface of the front contact of the key, closes the secondary circuit, and allows the charges to be sent into the roof, while, when the key is on the back contact, the secondary and primary coils are cut out, and the charge from the roof then passes direct to the key and through the telephone to earth, which, as a rule, is made by connecting wires from the coil and the telephone to one of the axle-boxes. The coil and the key, with suitable connections, are mounted on a board which is large enough to contain a telegraph form besides, and the telephone is attached by flexible connections, and is, when in use, strapped to the operator's head. The battery is put up in a case with a handle, so that the whole apparatus can be carried from one end of a train to the other. The arrangements at the terminal and other stations on the line, so far as induction telegraphy is concerned, are practically identical with those in the railway coach ; but, in addition, they have a duplex Morse equipment, by which ordinary messages can be sent by the dot-and-dash system.

Of late years interesting experiments have been made under the direction of Mr. W. H. Preece in carrying on telegraphic communication across considerable distances by means of induction between parallel circuits.

§ 4. Induced Currents of Higher Orders.— In 1888 Henry made a further remarkable discovery, viz., that secondary currents, though only of momentary duration, could in their turn induce other induced currents in neighbouring conductors ; and these he called tertiary and currents of higher orders.

A primary current was passed through coil No. 1, while coil No. 2 was placed over it to receive the secondary current, and the ends of this last coil joined to a third coil, No. 3. By this disposition the secondary current passed through No. 3, and sinca this was at a distance (see Fig. 81), and beyond the influence of the primary, its separate induction could be rendered manifest by the effects on helix No. 1, arranged as a secondary circuit to this third coil. When the handles a ft of the last helix d were grasped a powerful shock was received, proving the induction of a tertiary current in the last

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MUTUAL AND SELF INDUCTION.

coil. By a similar more extended arrangement of inducing coils (shown in Fig. 82) shocks were received from currents of a fourth and fifth order; and with a more powerful primary current and additional coils a still greater number of successive inductions might be obtained. Henry thus established by decisive experiments that, in a properly placed series of connected coils, a primary current could give birth to secondary currents, and these last to tertiary currents, and

FIG. 81.

so on, a whole family of induced currents arising from the starting or stopping of the primary current.

It was found that with a small battery a shock could be given from the current of the third order to 25 persons joining hands ; also shocks perceptible in the arms were obtained from a current of the fifth order.

FIG. 82.

When the long helix is placed over a secondary current generated in a short coil, and which is one of quantity, a tertiary current of intensity was obtained capable of producing shocks. When the intensity current of the last experiment was passed through a second helix, and another flat riband coil placed over this (see Fig. 82 A), a quantity current was again produced. Therefore, in the case of these currents of higher orders also a quantity current could be induced from one of intensity, and the converse.

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The arrangement in Fig. 82 shows these different results produced at once. The induction from coil No. 3 to helix No. 1 produces an intensity current, and from helix No. 2 to coil No. 4 a quantity current.

The next stage in Henry's inquiry had reference to the direction of these induced currents. Knowing that a current on starting in a conductor induces an inverse or oppositely- directed induced current in a neighbouring secondary circuit, and a direct or like directed induced current on stopping, it was clear that each tertiary current must consist, in its simplest form, of two oppositely directed currents succeeding each other instantaneously ; for at the " make " or "break" of the primary the secondary circuit is traversed by a brief secondary current in "opposite" or "like" direction. We shall speak of these as the inverse and direct secondary currents produced on closing or opening the primary circuit.

FIG. E2.v.

Each of these secondary currents rises to a maximum and then sinks to zero again. If there is a tertiary circuit present, then during the rise of the secondary current to its maximum it is developing an inverse tertiary and during its decrease to zero a direct tertiary current. Since, as we shall see, the duration of the secondary current is a very small fraction of a second, these two component tertiary currents must succeed each other at an excessively short interval of time. Physio- logically their separate effect is, so to speak, united, and they make themselves felt as one shock. Henry adopted the method of employing a magnetising spiral containing a sewing needle as a means of analysing the nature of these induced currents of higher order. By inserting such a spiral in the circuit of the successive conductors and noting the direction of the magnetisation of the steel needle he arrived at the conclusion that there exists an alternation in

222 MUTUAL AND SELF INDUCTION.

the direction of the currents of the several orders, and that the directions of the several induced currents could he expressed by saying that at the " make " of the primary we get an inverse secondary, a direct tertiary, and inverse •quarternary current, and so on ; or, symbolically : — Primary current started stopped

Secondary ,, inverse — direct -

Tertiary ,, direct + inverse -j-

Quarternary „ inverse — direct —

d-c. dc.

The use of a magnetising spiral as a means of determining the direction of an induced current is, however, liable to lead to serious errors in drawing conclusions as to direction of currents, and the above experiment cannot be regarded as an exhaustive examination of the whole phenomena of induced currents of higher orders. Before entering into a more detailed discussion of the exact nature of the effects which here present themselves, it will be of assistance to gather together the principal observations on the induction of transient electric currents.

§ 5. Inductive Effects Produced by Transient Electric Cur- rents.— It was an obvious inference, from all the foregoing facts, that Leyden jar discharges, or the transient currents formed by discharging charged condensers, should in like manner be able to give rise to a family of induced currents in suitably-placed circuits. Henry thus opened up a new field of research, which was diligently cultivated by Marianini, Abria, Matteucci, Eeiss, Verdet, and many other physicists. Henry's first experiment was as follows : A hollow glass cylinder (see Fig. 83) of about six inches hi diameter was prepared with a narrow riband of tinfoil about thirty feet long pasted spirally around the outside, and a riband of the same length pasted on the inside, so that the corresponding spires of the two were directly opposite each other. The ends of the inner spiral passed out of the cylinder through a glass tube to prevent direct communication between the two circuits. When the ends of the inner riband were joined by the magnetising spiral containing a sewing needle and a discharge from a half-gallon jar sent through the outer riband, the needle was strongly

MUTUAL AND SELF INDUCTION. 223

magnetised in such a manner as to indicate an induced current through the inner riband in the same direction as that of the current of the jar. If, instead of using the magnetising spiral, the ends of the inner riband were brought near together, a small spark was detected at the instant of sending a jar dis- charge through the outer conductor. Experiments were next made in reference to the production of induced currents of different orders by electric discharges. For this purpose a series of glass cylinders with tinfoil spirals pasted on them was prepared and joined up so that the inner spiral of one cylinder was in connection with the outer spiral of another. When a discharge was passed through the outer riband of the first cylinder it produced an induced secondary discharge circulating in the inner spiral of the first and the outer spiral of the second cylinder. This in turn generated a tertiary current, and so

FIG. 83.

forth. Each of these discharges was a brief wave of current, and by the use of the magnetising spiral in each circuit an attempt was made to determine the direction of the discharge. Here, however, an anomaly presented itself. By the use of this magnetising spiral it appeared that the induced discharges were all in the same direction. Leyden jar discharges were then passed through the first member of the series of coils and helices used in the experiments on galvanic currents, and here the directions of the induced discharges in the several con- ductors were found to alternate. After various experiments Henry considered that he had found the solution of this anomaly in the different distances, of the inducing and induc- tive circuits. As an experiment illustrating this he gives the following: — Two narrow strips of tinfoil about twelve feet long

224 MUTUAL AND SELF INDUCTION.

were stretched parallel to each other, and separated by thin plates of mica to the distanca of about ^th of an inch. When a discharge from a half-gallon jar was passed through one of these an induced current in the same direction was obtained from the other. When the ribands were separated to a distance of about ith of an inch, no induced current, as evidenced by the absence of effect in the magnetising spiral, could be obtained. When the circuits were still further separated the induced current reappeared, but in the opposite direction to the primary discharge. The distance at which the induced dis- charge changes direction appears, according to Henry, to be dependent on a number of circumstances, such as the capacity and charge of the jar and the length and thickness of the wires.

With a battery of eight half-gallon jars and parallel wires of about ten feet long the change in direction did not take place until the wires were separated by twelve or fifteen inches. The currents of all the higher orders were found to change sign with a change in the distance between the inducing and inductive circuit.

One interesting experiment was made by Henry to illustrate the inductive effect of jar discharges across considerable dis- tances. In this case a primary circuit was formed consisting of an insulated wire eighty feet long. Around this, and separated from it by a distance of about twelve feet, was another circuit consisting of a wire one hundred and twenty- feet long. When the discharge from thirty large Leyden jars was sent through the primary wire an induced discharge was obtained in the other sufficiently strong to magnetise to saturation a small needle placed in a magnetising spiral interpolated in the secondary circuit. We may, however, remark here, once for all, that all these experiments directed to determine the direction of induced discharges in which the magnetising power of the discharge is made use of for this purpose are difficult to interpret, and too much reliance must not be placed on the conclusions thus drawn. Leaving out of account for the moment all consideration of what are called electric oscillations, to which we shall allude subsequently, we may say that, if two discharges are passed through a magnetis- ing spiral, the discharges being oppositely directed and of equal

MUTUAL AND SELF INDUCTION. 225

quantity but different durations, the resulting direction of magnetisation will be dependent upon several conflicting elements. Speaking generally, the intensity of magnetisation is determined by the relative magnitude of the maximum current strength during the discharge, and of two discharges having equal quantity the one lasting the shortest time would rise to the highest current strength during the period of the discharge, and exercise the greatest magnetising force. Even then it would not be safe to draw too pronounced a con- clusion from the direction of magnetisation as to the relative magnitudes of the maxima of two alternate discharge currents rapidly succeeding each other, for, as Abria pointed out long ago,* the demagnetisation of a steel needle requires a less magnetising force than that necessary to magnetise it in the first instance, and hence the final results are complicated by the relative order of imposition as well as the relative maximum magnitude of the magnetising discharge currents. One fact which has to be borne in mind in attempting to interpret these results of Henry is that the magnetising current whose direction we are seeking to determine acts by induction also on the mass of the needle or iron in the testing magnetising coil, and generates in its mass induced currents circulating round its surface. Under the head of Magnetic Screening in a later section we shall examine the circumstances under which such currents induced in a metallic mass shield to a greater or less extent conducting circuits lying beyond them from inductive effects. Meanwhile we may say that the effect of a very sudden discharge in one direction in the magnetising coil is to induce eddy currents in the surface of the needle which shield the inner and deeper portions of the steel from the magnetising action, and the resulting magnetisation is chiefly superficial. If, however, the dis- charge is prolonged or dragged out whilst retaining the same electric quantity, the shielding action will not be so pronounced, and the magnetisation will penetrate deeper down into the mass of the steel. Accordingly two equal discharges, i.e., discharges of equal quantity, may produce a greater or less magnetic moment in the steel, according as the duration of the same is greater or less, a very sudden discharge

• Abria, Ann. dc Chcm. et dc Phys., [3] Vol. I., p. 429, 1844.

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226 MUTUAL AND SELF INDUCTION.

Laving much less magnetic-moment -producing power than the same quantity more dragged out. We may in general also say that the magnetising power of a discharge current is determined by the value of the maximum current strength during the discharge, and hence of two equal quantity dis- charges, the one which lasts the shorter time, and which has, therefore, the greatest maximum value, will, if the discharges are approximately equal in duration, produce the greatest magnetising effect.

The tertiary currents, produced by ordinary galvanic currents, and the secondary currents, produced by Ley den jar discharges, consist, as we have seen, in their simplest form of a double discharge or flow, one part inverse, or oppositely directed to its inducing parent current, and the succeeding part direct, or similarly directed, the two component currents of the total discharge having equal quantity but different durations. In general the first portion, or the inverse current, is that which has the greatest maximum value and the shortest duration, the second half, or the direct current, being more dragged out in tune ; and, for a reason to be stated further on, the approxima- tion of the induced and inducing circuits exaggerates this difference, or increases the maximum value of the inverse current at the expense of its duration. The explanation which may be offered, however, of the phenomena of the magnetisa- tion of steel by tertiary currents, or by the secondary currents due to Leyden jar discharges, is as follows : — When the induced and inducing circuits are not very near to each other, and when the inducing current reaches its maximum not very suddenly, the two induced currents are not very different in duration, but the first or inverse current has, of the two, a rather greater maximum and less duration. It follows that a magnetisa- tion is produced in the needle, which, on the whole, is in the direction produced by the inverse current, and the inference from the direction of magnetisation is that the induced and inducing currents are in the opposite direction. If, however, the inducing current reaches its maximum value very suddenly, as it does if the circuits are very close, then the first half, or the inversed induced current, is so brief in its duration that the magnetisation of the needle due to it is very superficial. On the other hand, the magnetisation due to the rather more pro-

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longed direct current is more diffused through the needle, and the resultant magnetisation found on testing the needle is that apparently due to the direct current, and the inference from the resulting magnetism of the needle would be that the induced and inducing currents are in the same direction. By some such explanation as the above we may reconcile these experimental results of Henry with known facts, but it is evident, since the resulting magnetisation of the needle is an effect determined by the relative maximum values of the two portions of the total induced current, and by their duration, as •well as by their order of superposition, that considerable caution is necessary in attempting to interpret the results of experi- ments made with a magnetising helix. Henry was followed in the same field of investigation by Abria, Marianini, Eeiss, and Matteucci. Matteucci endeavoured to determine the direction of the induced discharges by employing a pro- cess founded upon the experiment of the pierced card, in which the hole made by the spark on a piece of paper or a card is always nearer to the negative electrode. By means of this process, combined with the employment of the galvanometer, Matteucci considered that the inductive discharges are deter- mined by the following law: — If the inducing and induced circuit are both closed, the induced discharge is in the opposite direction to the inducing discharge. If, however, the induced circuit is interrupted at any point so that there is a spark, the induced discharge is in the same direction as the inducing. Abandoning these methods above described, M. Verdet* em- ployed another, which depends upon the polarisation of elec- trodes in dilute sulphuric acid.

From more recent knowledge we may state the facts with regard to the action of alternate currents upon a dilute sul- phuric acid voltameter as follows! : —

If a current of electricity consisting of alternate short fluxes of current of opposite sign is passed through a voltameter having platinum electrodes, and if these electrodes are large, there is no visible decomposition, but if the electrodes are reduced in size below a certain limit visible decomposition begins. For every

• Verdet, An*, de Chem. et de Phys., [3] Vol. XXIX., p. 501, 1850. t See a Paper by MM. Maneuvrier and J. Chappuis, abstracted in The Electrician, June 29, 1888, Vol. XXL, p. 237.

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228 MUTUAL AND SELF INDUCTION.

current there is a certain size of electrode, above which gas is not visibly evolved, and for every given size of electrode there is a current below which gas is not apparently liberated. When the conditions are suitable for the liberation of gas, the gases collected at both electrodes have the same composition. If the quantities of electricity passing in each alternate and oppositely- directed flux are equal, then the electrodes are not sensibly polarised. If, however, the quantities are not equal, then there is, on the whole, a greater flow of current in one direction than in the other, and the electrodes exhibit the state known as polarisation, and yield a reverse current when connected with the galvanometer. Verdet, in his experiments, made use of flat spirals, the wires of which were insulated from each other with great care by silk and a layer of gum-lac varnish. The primary spiral was made of copper wire fths of an inch in dia- meter and 92 feet in length, forming 24 spirals. The secondary circuit consisted of three spirals of wire -^th of an inch in dia- meter and 157 feet in length, making 95 turns. The inducing discharge was supplied from a Leyden jar battery of nine large jars. The induced discharge was sent through a voltameter having small platinum electrodes, and which could be connected •with a delicate galvanometer for detecting polarisation of the electrodes immediately after the discharge. Verdet's experi- ments led him to recognise that when the induced circuit is continuous, and not interrupted anywhere except by the insertion of the voltameter, no traces of polarisation are obtained except by very powerful discharges. This indicates that the induced discharge consists of a double current of two oppositely- directed and equal quantities of electricity. In the case of very powerful discharges there was a slight galvano- metric deflection, indicating a preponderating secondary dis- charge in the same direction as the primary. If the induced or secondary circuit is interrupted at one point, so that the discharge has to pass as a spark at that place, then very per- ceptible polarisation of the electrodes presents itself, and the direction of this is such as to indicate a predominant induced current passing in the same direction as the primary.

To sum up. It follows from all the numerous researches on induced discharges that this is a very complex phenomenon, and is influenced by a large number of conditional circum-

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stances, and also by the very mode employed for determining it. It may be, however, taken as proved that an induced dis- charge, produced either as a secondary discharge by a transi-' tory primary, such as the discharge from a Leyden jar, or a tertiary current produced by induction by a secondary current of very brief duration, is, hi its simplest form, a wave of electric current, consisting of two short fluxes or currents in opposite directions, and succeeding each other immediately. This Poggendorff* holds to be shown by the action of such tertiary or higher order currents on a galvanometer. If these currents are led through a galvanometer of which the arrangement is such that the magnetic axis of the needle is accurately at right angles to the direction of the magnetic axis of the coil, then no deflection of the needle is observed, or at most a very slight •one. If, however, the needle makes an angle with the plane of the coils, then these induction currents cause a deviation of the needle. This effect (die doppehinnige Ablenkung) arises from the fact that the magnetism of the needle is not rigid, and that the alternate twisting couples to which the needle is subjected are not equal, by reason of the fact that one of the halves of the complete induced current — say the direct half— increases the magnetic moment of the needle, and hence increases slightly the deflecting couple in the direction tending to increase the deviation of the needle ; the other half — say the inverse part of the induced current — tends to reduce the moment of the needle, and hence to subject it to a smaller reverse couple. Hence it follows that, if discharges of equal quantity and opposite sign succeed each other through a galvanometer when the needle is accurately in the plane of the coils, Little or no deviation is observed ; but if the coils are turned so that the needle makes an angle with them, then these alternate currents' will affect the needle and increase the ;,angle of deflection.

This behaviour towards a galvanometer, and the action on a voltameter of liberating mixed gases of equal composition at each pole, prove that each induced current of the third and higher orders consists of two oppositely-directed discharges, produced by the operation of two successive electromotive impulses of opposite sign and very brief duration acting upon

• Poggendorff, Pogg. Ann,, Bd. XLV., 1838, p. 353.

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the circuit. The quantities of these discharges are equal ; but the durations are different, and hence the maximum value of the current strength during the opposite discharges may be very different.

This may be illustrated graphically thus : —

Let the curve a P b Q c (Fig. 84) be a current curve represent- ing two waves of current of opposite sign succeeding each other. Let the horizontal line a c be a time line, and vertical ordinates represent instantaneous current strengths. Then the shaded areas will represent the quantity in each discharge. Let these shaded areas be equal, then the diagram represents two discharges of equal quantity succeeding each other in opposite directions, but having different maximum current

FIG. 84.

strengths I and I'. The duration of the first discharge is represented by a b, and that of the second by b c. This diagram represents the conditions in the simplest case of tertiary current. If the instantaneous value of the current at any time is called i, then the whole quantity of the discharge will be represented by the shaded area and by the integral / i d t between proper limits.

We may classify the effects of induced discharges or currents in the following way : —

(1) Those effects dependent upon I i d t, or upon the whole quantity of the discharge. These are the gahanometric and the electro-chemical effects. If a discharge is passed through a.

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galvanometer, the duration of which is very small compared with the time of free oscillation of the needle, the galvanometer needle experiences a "throw" such that the sine of half the angle of deflection is proportional to the whole quantity of the discharge. Also in a voltameter, by Faraday's law, the whole quantity of the electrolyte broken up is proportional to the quantity of electricity which has passed through it.

(2) Those effects dependent upon I i-dt, or upon the average of the square of the strength of the current at every instant during the discharge. These are the heating and the electro- dynamic effects. By Joule's law, at every instant the rate of dissipation of energy is proportional to the square of the current strength, and hence the whole heat generated by the discharge is proportional to the integral above. Similarly, if the dis- charge passes through a circuit, part of which is movable and can react upon a fixed part, so that attraction or repulsion may take place between them, the force is dependent at any instant on the square of the current strength, and hence the whole effect or average force upon the same integral.