The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 25 of 35

As the induced sparks in the experiment last described were several millimatres in length, Hertz modified it by using the arrangement shown in Fig. 154, and greatly increasing the distance between the micrometer circuit and the secondary circuit of the induction coil. The terminal conductors C and C'' were three metres apart, and the wire between them was of copper, 2 millimetres in diameter, with the discharger B at its centre.

The micrometer circuit consisted, as in the preceding experi- ments, of a rectangle 80 centimetres broad by 120 centimetres long. With the nearest side cf the micrometer circuit at a

DYNAMICAL THEORY OF INDUCTION. 427

distance of half a millimetre from C B C', sparks two milli- metres in length were obtained at M, and though the length of the sparks decreased rapidly as the distance of the micrometer circuit was increased, a continuous stream of sparks was still obtained at a distance of one and a-half metres. The interven- tion of the observer's body between the micrometer circuit and the wire C B C' produced no visible effect on the stream of sparks at M. That the effect was really due to the rectilinear conductor C B (7 was proved by the fact that when one or other, or both, halves of this conductor were removed, the sparks at M ceased. The same effect was produced by drawing the knobs of the discharger B apart until sparks ceased to pass, showing that the effect was not due to the electrostatic potential differ- ence of C and C', as this would be increased by separating the discharger knobs beyond sparking distance.

The closed micrometer circuit was then replaced by a straight copper wire, slightly shorter than the distance C C', placed parallel to C B C' and at a distance of GO centimetres from it. This wire terminated in knobs, 10 centimetres in diameter, attached to insulating supports, and the spark micrometer divided it into two equal parts. Under these circumstances sparks were obtained at the micrometer as before.

With the rectilinear open micrometer circuit sparks were still observed at the micrometer when the discharger knobs of the secondary coil circuit were separated beyond sparking distance. This was, of course, due simply to electrostatic induction, and shows that the oscillatory current in C C' was superposed upon the ordinary discharges. The electrostatic action could be got rid of by joining the micrometer knobs by means of a damp thread. The conductivity of this thread was therefore sufficient to afford a passage to the comparatively slow alternations of the coil discharge, but was not sufficient to provide a passage for the immeasurably more rapid alternations of the oscillatory current. Considerable sparking took place at the micrometer when its distance from C B C' was 1'2 metre, and faint sparks were distinguishable up to 3 metres. At these distances it was not necessary to use the damp thread to get rid of the electrostatic action, as, owing to its diminishing more rapidly with increase of distance than the effect of the current induction, it was no longer able to produce sparks in the micro-

423 D YNA MICAL THEOR Y OF IND UCTION.

meter, as was proved by separating the discharger knobs beyond sparking distance, when sparks could no longer be perceived at the micrometer.

Resonance Phenomena. — In order to determine whether the oscillations were of the nature of a regular vibration, Hertz availed himself of the principle of resonance. According to this principle, an oscillatory current of definite period would, other conditions being the same, exert a much greater inductive effect upon one of equal period than upon one differing even slightly from it.*

If, then, two circuits are taken having as nearly as possible equal vibration periods, the effect of one upon the other will be diminished by altering either the capacity or the coefficient of self-induction of one of them, as a change in either of them would alter the period of vibration of the circuit.

This was carried out by means of an arrangement very simi- lar to that of Fig. 154. The conductor C C' was replaced by a .straight copper wire 2-6 metres in length and 5 millimetres in diameter, divided into two equal parts as before by a discharger. The discharger knobs were attached directly to the secondary terminals of the induction coil. Two hollow zinc spheres, 30 centimetres . in diameter, were made to slide on the wire, one on each side of the discharger, and since, electrically speaking, these formed the terminals of the conductor, its length could be varied by altering their position. The micrometer circuit was chosen of such dimensions as to have, if the author's hypothesis were correct, a slightly shorter vibration period than that of C C'. It was formed of a square, with sides 75 centi- metres in length, of copper wire 2 millimetres in diameter, and it was placed with its nearest side parallel to CB C' and at a distance of 30 centimetres from it. The sparking distance at the micrometer was then found to be 0-9 millimetre. When the terminals of the micrometer circuit was placed in contact with two metal spheres 8 centimetres in diameter, supported on insulating stands, the sparking distance could be increased up to 2-5 millimetres. When these were replaced by much larger spheres the sparking distance was diminished to a small fraction of a aaillimetre. Similar results were obtained on .connecting the micrometer terminals with the plates of a

• See Oberbeck, Wiederuanu's Annalen, Vol. XXVI., p, 245, 1835.

DYNAMICAL IHEORY OF INDUCTION.

429

Kohlrausch condenser. When the plates were far apart the increase of capacity increased the sparking distance, but when the plates were brought close together the sparking distances- again fell to a very small value.

The simplest method of adjusting the capacity of the micro- meter circuit is to suspend to its ends two parallel wires the distance and lengths of which are capable of variation. By this means the author succeeded in increasing the sparking distance up to three millimetres, after which it diminished when th& wires were either lengthened or shortened. The decrease of the sparking distance on increasing the capacity was naturally to be expected ; but it would be difficult to understand, except on the principle of resonance, why a decrease of the capacity should have the same effect.

FIG. 155.— Curve showing relation between length of side of rectangle (taken as abscissa) and maximum sparking distance (taken as ordinate), the- sides consisting of straight wires of varying lengths.

The experiments were then varied by diminishing the capa- city of the circuit C B C' so as to shorten its period of oscil- lation, and the results confirmed those previously obtained . and a series of experiments in which the lengths and capacities of the circuits were varied in different ways showed conclu- sively that the maximum effect does not depend on the con- ditions of either one of the two circuits, but on the existence of the proper relation between them.

When the two circuits were brought very close together, and the discharger knobs separated by an interval of 7 millimetres, sparks were obtained at the micrometer, which were also

430

DYNAMICAL THEORY OF INDUCTION.

7 millimetres in length, when the two circuits had been care- fully adjusted to have the same period. The induced E.M.F's must in this case have attained nearly as high a value as the inducing ones.

To show the effect of varying the coefficient of self-induction, a series of rectangles, abed (Fig. 154), were taken, having a constant breadth, a b, but a length, a c, continually increasing from 10 centimetres up to 250 centimetres : it was found that the maximum effect was obtained with a length of 1-8 metre. The quantitative results of these experiments are shown in Fig. 155, in which the abscissae of the curve are the double lengths of the rectangles, and the ordinates represent the cor. responding maximum sparking distances. The sparking dis- tances could not be determined with great exactness, but the

• FIG. 156. — Curve showing relation between length of side of rectangle •(taken as abscissa) and maximum sparking distance (taken as ordinate), the sides consisting of spirals gradually drawn out.

errors were not sufficient to mask the general nature of the result.

In a second series of experiments the sides a c and b d were formed of loose coils of wire which were gradually pulled out, and the result is shown in Fig. 156. It will be seen that the maximum sparking distance was attained for .a somewhat greater length of side, which is explained by the fact that in the latter experiments the self-induction only was increased by increase of length, while in the former series the capacity was increased as well. Varying the resistance of the micrometer

DYNAMICAL THEORY OF INDUCTION. 431

circuit by using copper and German silver wires of various diameters was found to have no effect on the period of oscilla- tion, and extremely little on the sparking distance.

When the wire c d was surrounded by an iron tube, oi when it was replaced by an iron wire, no perceptible effect was obtained, confirming the conclusion previously arrived at that the magnetism of the iron is unable to follow such rapid oscillations, and therefore exerts no appreciable effect.

It is only proper, however, to interpolate at this point the remark that other observers do not endorse entirely this statement of Hertz. We may especially draw attention to the work of Prof. J. Trowbridge and of Mr. C. E. St. John* on the propagation of electrical oscillations on iron wires. The experimental results obtained by these investi- gators may be summed up as follows : —

1. The magnetic permeability of iron wires exercises an Important influence upon the decay of electrical oscillations of high frequency. The influence is so great that the oscilla- tions may be reduced to half an oscillation on a circuit of suitable self-induction and capacity for producing them.

2. Currents of high frequency such as are produced in Leyden jar discharges therefore magnetise iron.

3. The self-induction of iron circuits is sensibly greater than that of similar copper circuits under rapid electrical oscillations 115 x 106 reversals per second.

4. This increase in self-induction produces a shortening of the wave-length.

o. The permeability of annealed iron under the atove rate of alternation is about 885.

For full information as to the methods of obtaining these results we must refer the reader to the original Papers.

Nodes. — The vibrations in the micrometer circuit which have been considered are the simplest ones possible, but not the only ones. While the potential at the ends alternates between two fixed limits, that at the central portion of the circuit retains a constant mean value. The electrical vibration, therefore, has

• See Phil. May., December, 1891, Mr. J. Trowbridge on " Damping of Electrical Oscillations on Iron Wires ; " and Phil. Mag., Xovember, 1894, Mr. C. E. St. John on " Wave-Lengths of Electricity on Iron Wires. '

432 DYNAMICAL THEORY OF INDUCTION.

a node at the centre, and this will be the only nodal point. Its- existence may be proved by placing a small insulated sphere close to various portions of the micrometer circuit while sparks are passing at the discharger of the coil, when it will be found that if the sphere is placed close to the centre of the circuit the sparking will be very slight, increasing as the sphere is moved further away. The sparking cannot, however, be entirely got rid of, and there is a better way of determining the existence and position of the node. After adjusting the two circuits to unison, and drawing the micrometer terminals so far apart that sparks can only be made to pass by means of resonant action, let different parts of the circuit be touched by a conductor of some capacity, when it will be found that the sparks disappear, owing to interference with the resonant action, except when the point of contact is at the centre of the- circuit. Hertz then endeavoured to produce a vibration with two nodes, and for this purpose he modified the apparatus previously used by adding to the micrometer circuit a second rectangle, ef g h, exactly similar to the first (as shown in Fig. 157), and joining the points of the circuit near the terminals by wires 1 3 and 2 4, as shown in the diagram.

The whole system then formed a closed metallic circuit, the fundamental vibration of which would have two nodes. Since the period of this vibration would necessarily agree closely with that of each half of the circuit, and, therefore, with that of the circuit C C', it was to be expected that the vibration would have a pair of loops at the junctions 1 3 and 2 4, and a pair of nodes at the middle points of c d and g h. The vibra- tions were determined by measuring the sparking distance between the micrometer terminals 1 and 2. It was found that, contrary to what was expected, the addition of the second rectangle diminished this sparking distance from about three millimetres to about one millimetre. The existence of resonant action between the circuit C C' and the micrometer circuit was, however, fully demonstrated, for any alteration in the circuit ef g h, whether it consisted in increasing or in decreasing its length, diminished the sparking distance. It was also found that much weaker sparking took place between c d or g h and an insulated sphere than between a e or bf and the same sphere, showing that the nodes were in c d and g hf

DYNAMICAL THEORY OF INDUCTION.

433

as expected. Further, when the sphere was made to touch c d or g h it had no effect on the sparking distance of 1 and 2 ; but when the point of contact was at any other portion of the circuit the sparking distance was diminished, showing that these nodes did really helong to the vibration, the resonant action of which increased this sparking distance.

The wire joining the points 2 and 4 was then removed. As the strength of the induced oscillatory current should be zero at these points, the removal ought not to disturb the vibrations, and this was shown experimentally to be the case, the resonant effects and the position of the nodes remaining unchanged. The vibration with two nodal points was, of

-0-

€)-

FIG. 157.

course, not the fundamental vibration of the circuit, which consisted of a vibration with a node between a and e, and for which the highest values of the potential were at the points 2 and 4.

When these spheres forming the terminals at these points were brought close together slight sparking was found to take place between them, which was attributed to the excitation, though only to a small extent, of the fundamental vibration. This explanation was confirmed in the following manner : — The sparks between I and 2 were broken off, leaving only the

434 DYNAMICAL THEORY OF INDUCTION.

sparks between 2 and 4, which measured the intensity of the fundamental vibration. The period of vibration of the circuit C C' was then increased by drawing it out to its full length, and thereby increasing its capacity, when it was observed that the sparking gradually increased to a maximum, and then began to diminish again. The maximum value must evidently occur when the period of vibration of the circuit C C' is the same as that of the fundamental vibration of the micrometer circuit, and it was shown that when the sparking distance between 2 and 4 had its maximum value the sparks corre- sponded to a vibration with only one nodal point, for the sparks ceased when the previously existing nodes were touched by a conductor, and the only point where contact could take place without effect on the sparking was between a and e. These results show that it is possible to excite at mil in the same conductor either the fundamental vibration or its first overtone, to use the language of acoustics.

Hertz appeared to consider it very doubtful whether it was possible to get higher overtones of electrical vibration, the •difficulty of obtaining such lying not only in the method of observation, but also in the nature of the oscillations them- selves. The intensity of these is found to vary considerably during a series of discharges from the coil even when all the circumstances are maintained as constant as possible, and the comparative feebleness of the resonant effects shows that there must be a considerable amount of damping. There are, more- over, many secondary phenomena which seem to indicate that irregular vibrations are superposed upon the regular ones, as would be expected in complex systems of conductors. If, therefore, we wish to compare electrical oscillations from a mathematical point of view with those of acoustics, we must seek our analogy in the high notes intermixed with irregular vibrations, obtained, say, by striking a wooden rod with a liammer rather than in the comparatively slow harmonic vibration of tuning forks or strings ; and in the case of vibra- tions of the former class we have to be contented even in the study of acoustics with little more than indications of such phenomena as resonance and nodal points.

Referring to the conditions to be fulfilled in order to obtain the best results, Hertz noted a fact of very considerable interest

DYNAMICAL THEORY OF INDUCTION. 435

and novelty, namely, that the spark from the discharger should always be visible from the micrometer, as, when this was not the case, though the phenomena observed were of the same character, the sparking distance was invariably diminished. This effect of the light from the spark of an induction coil in increasing the sparking distance in a secondary circuit has been fully described by Dr. Lodge in his book on the work of Hertz, and he has pointed out that the same effect is produced by light from burning magnesium wire or other sources rich in the ultra-violet rays.

Theory of the Experiments. — The theories of electrical oscilla- tions which have been developed by Lord Kelvin, von Helmholtz, and Kirchoff have been shown* to hold good for the open circuit oscillations of induction apparatus, as weh1 as for the oscillatory Leyden jar discharge ; and it is of interest to inquire whether the observed results are of the same order as those indicated by theory.

Hertz considers, in the first place, the vibration period. Let T be the period of a single or half vibration proper to the conductor exciting the micrometer circuit ; L its coefficient of self-induction in absolute electromagnetic measure, expressed, therefore, in centimetres ; C the capacity of one of its terminals in electrostatic measure, and therefore also expressed in centi- metres ; and v the velocity of light in centimetre-seconds ; then, if the resistance of the conductor is small,

v

In the case of the resonance experiments, the capacity C was approximately the radius of the sphere forming the terminal, so thr.t C = 7'5 centimetres.! The coefficient of self-induction

• Lorentz, Wiedemann's Annalen, Vol. VII., p. 161, 1879.

t In Hertz's original Paper the capacity of the spherical terminal ball was taken as 15 units. M. Poincare first drew attention to the fact that the capacity C in the above formula denotes the amount of electricity which exists at one end of an oscillating conductor when the difference of potential between the two ends is equal to unity. Hence, if the spheres are far apart, the difference of potential between each of them and sur- rounding space is +|. Therefore the charge on the sphere is formed by dividing its capacity, i.e., its radius in centimetres, by 2. Hence, C in the

FF2

436 DYNAMICAL THEORY OF INDUCTION.

was that of a wire of length 1 = 150 centimetres and diameter d = 1/2 centimetre.

According to Neumann's formula,

Lf [cos € , j , = I I ds dst

which gives in the case considered

L = 2 7 (log ^L - 0-75^ = 1,902 centimetres. \ d /

As, however, it is not quite certain that Neumann's formula is applicable to an open circuit, it is better to use von Helm- holtz's more general formula, containing an undetermined constant k, according to which

Putting Jc = l, this reduces to Neumann's formula; for & = 0< it reduces to that of Maxwell, and for A- = — 1 to Weber's. The greatest difference in the values of L obtained by giving these different values to k would not exceed a sixth of its mean value, and therefore, for the purposes of the present approxi- mation, it is enough to assume that k is not a large positive or negative number; for if the number 1,902 does not give the correct value of the coefficient for the wire 150 centimetres in length, it will give the value corresponding to a conductor not differing greatly from it in length.

Taking L = 1,902 centimetres, we have TT VC L = 531 centi- metres, which represents the distance traversed by light during the oscillation, or, according to Maxwell's theory, the length of an electromagnetic ether wave. The value of T is then found to be 1-26 hundred-millionths of a second, which is of the same order as the observed results.

The ratio of damping is then considered. In order that oscillations may be possible, the resistance of the open circuit must be less than 2 v /L/C. Fcr the exciting circuit used this gives 676 ohms as the upper limit of resistance. If the actual resistance, r, is sensibly below this limit, the ratio of damping

will be e*t. The amplitude will therefore be reduced in the ratio 1:2-71 in

2L_2j,- /L 67G 215 rT"»rV C"»r".r

D YNA MICA L THEOE Y OF IND UCTION. 437

oscillations. We have, unfortunately, no means of deter- mining the resistance of the air space traversed by the spark, but as the resistance of a strong electric arc is never less than a few ohms we shall be justified in assuming this as the minimum limit. From this it would follow that the number of oscillations due to a single impulse must be reckoned ill tens, and not in hundreds or thousands, which is in accordance with the character of the experimental results, and agrees with results observed in the case of the oscillatory Leyden jar discharge. In the case of closed metallic circuits, on the other hand, theory indicates that the number of oscillations before equilibrium is attained must be reckoned by thousands. Hertz compares, lastly, the order of the inductive actions of these oscillations according to theory with that of the effects actually observed. To do this it must be noted that the maximum E.M.F. induced by the oscillation in its own circuit is approximately equal to the maximum potential difference at its extremities ; for if there were no damping these quanti- ties would be identical, since at any moment the potential difference at the extremities and the E.M.F. of induction would be in equilibrium. In the experiments under con- sideration the potential difference at the extremities was such as to give a spark 7 to 8 millimetres in length, which must therefore represent the maximum inductive action excited in its own circuit by the oscillation. Again, at any instant the induced E.M.F. in the micrometer circuit must be to that in the exciting conductor in the same ratio as that of the coefficient of mutual induction M of the two circuits to the coefficient of self-induction L of the exciting circuit. The value of M for the case considered is easily calculated from the ordinary formulae, and it is found to lie between one-ninth and one-twelth of L. This would only give sparks of from £ to f millimetre in length, so that according to theory visible sparks ought in any case to be obtained ; but, on the other hand, sparks several millimetres in length, as were obtained in the experiments previously described, can only be explained on the assumption that the successive inductive actions pro- duce an accumulative effect ; so that theory indicates the necessity of the existence of the resonant effects actually observed.

438 DYNAMICAL THEORY OF INDUCTION.

Hertz was at first inclined to suppose that as the micro- meter circuit was only broken by the extremely short air space limited by the maximum sparking distance under the condi- tions of the experiment, it might therefore be treated as a closed circuit, and only the total induction considered. The ordinary methods of electro-dynamics give the means of com- pletely determining the total inductive effect of a current element on a closed circuit, and would, therefore, in this case have sufficed for the investigation of the phenomena observed. He found, however, that the treatment of the micrometer circuit as a closed circuit led to incorrect results, so that it, as well as the primary, had to be treated as an open circuit, and therefore a knowledge of the total induction was insuffi- cient, and it became necessary to consider the value both of the E.M.F. of induction and of the electrostatic E.M.F. due to the charged extremities of the exciting circuit at each point of the micrometer circuit.

The investigations to which these considerations led are described by Hertz in a Paper, " On the Action of a Rectilineal- Electrical Oscillation upon a Circuit in its Vicinity," published in Wiedemann's Annalen, Vol. XXXIV., p. 155, 1883.

In what follows the exciting circuit will be spoken of as the primary and the micrometer circuit as the secondary. Hertz points out that the reason that electrostatic effect cannot be neglected is to be found in the extreme rapidity with which the electrostatic forces change their sign. If the electrostatic alternations in the primary were comparatively slow they might attain a very high intensity without giving rise to a spark in the secondary, since the electrostatic distribution on the secondary would vary so as to remain in equilibrium with the external E.M.F. This, however, is impossible, because the- variations in direction follow each other too rapidly for the distribution to foUow them.

In the present investigations the primary circuit consisted of a straight copper wire 5 millimetres in diameter, carrying at its extremities hollow zinc spheres 30 centimetres in diameter- The centres of the spheres were one metre apart, and at the middle of the wire was an air space f centimetre in length. The wire was placed in a horizontal position, and the observa- tions were all made at points near to the horizontal plane

DYNAMICAL THEORY OF INDUCTION. 439

through it, which, however, did not of course affect their generality, as the same effects would necessarily be produced in any plane through the horizontal wire. The secondary circuit consisted of a circle of 35 centimetres radius, of copper wire 2 millimetres in diameter, the circle being broken by an air space capable of variation by means of a micrometer screw.*

The circular form was selected for the secondary circuit because the former investigations had shown that the sparking distance was not the same at all points of the secondary, even when the conductor as a whole remained unchanged in posi- tion, and with a circular circuit it was easier to bring the air space to any part than if any other form had been used. To attain this object the circle was made movable about an axis passing through its centre perpendicular to its plane.

The circuits of the dimensions stated were very nearly in unison, and they were further adjusted by means of little strips of metal soldered to the extremities, and varied in length until the maximum sparking distance was obtained.

We shall follow Hertz in first considering the subject theoretically, and then examining how far the experimental results are in accordance with the theoretical conclusions. It will be assumed that the E.M.F. at every point is a simple harmonic function of the time, but that it does not undergo reversal in direction, and it will further be assumed that the oscillations are at any given moment everywhere in the same phase. This will certainly be the case in the immediate neigh- bourhood of the primary, and for the present we shall confine our attention to such points. Let s be the distance of a point measured along the circuit from the air space of the secondary, and F the component E.M.F. at that point along the circular arc d s. Then F is a function of s, which assumes its original value after passing once round the circle of circumference S. It may, therefore, be expanded in the form

JTTS B'sin27rs +

"s^ £ a IT

• This small circular detector circuit may be called an electro-magnetic eye, because it enables us to see the electro-magnetic disturbance and to localise it.

440 DYNAMICAL THEORY OF INDUCTION.

The higher terms of the series may he neglected, as the only result of so doing will be that the approximate theory will give an absolute disappearance of sparks where really the disappearance is not quite complete, and indeed the experi- ments are not delicate enough to enable us to compare their results with theory beyond a first approximation.

The force A acts in the same direction and is of constant amount at all points of the circle, and therefore it must be independent of the electrostatic E.M.F., as the integral of the latter round the circle is zero. A, then, represents the total E.M.F. of induction, which is measured by the rate of varia- tion of the number of magnetic lines of force which pass through the circle. If the electro-magnetic field containing the circle is assumed to be uniform, A will therefore be proportional to the component of the magnetic induction perpendicular to the plane of the secondary. It will therefore vanish when the direction of the magnetic induction lies in the plane of the secondary. A will consist of an oscillation, the intensity of which is independent of the position of the air space in the circle, and the corresponding spaiking dis- tance will be called a.

The term B' sin '\ can have no effect in exciting the funda- mental vibration of the secondary, since it is symmetrical on opposite sides of the air space. o_ „

The term B cos *~~ will give a force acting in the same

fc>

direction in the two quadrants opposed to the air space, and will excite the fundamental vibration. In the two quadrants adjacent to the air space it will give a force in the opposite direction, but its effect will be less than that of the former one ; for the current is zero at the extremities of the circuit, and therefore the electricity cannot move so freely as near the centre. This corresponds to the fact that if a string fastened at each end has its central portion and ends acted on respec- tively by oppositely-directed forces, its motion will be that due to the force at the central portion, which will excite the fundamental vibration if its oscillations are in unison with the latter. The intensity of the vibration will be proportional to B. Let E be the total E.M.F. in the uniform field of the

DYNAMICAL THEOET OF INDUCTION. 441

secondary, cf> the angle between its direction and the plane of •the latter, and 6 the angle which its projection on this plane makes with the radius drawn to the air space. Then we shall have, approximately,

S and, therefore, B = - E cos <£ sin 0.

B, therefore, is a function simply of the total E.M.F. due both to the electrostatic and electro -dynamic actions. It will vanish when $ = 90° — that is to say, when the total E.M.F. is perpendicular to the plane of the circle, whatever be the posi- tion of the air space on the circle. B will also vanish when 0 = 0 — that is to say, when the projection of the E.M.F. on the plane of the circle coincides with the radius through the air space. If the position of the air space on the circle is varied, the angle B will vary, and, therefore, also the intensity of the vibration and the sparking distance. The sparking distance corresponding to the second term of the expansion for F can therefore be represented approximately by a formula of the form (3 sin 9.

Now, the oscillations giving rise to sparks of lengths a and ,/3 sin 0 respectively are in the same phase. The resulting oscillations will therefore be in the same phase, and their amplitudes must be added together. The sparking distance being approximately proportional to the maximum total amplitude may therefore also be obtained by adding the sparking distances due to the two oscillations respectively. The sparking distance will therefore be given as a function of the position of the air space on the secondary circuit by the expression a + (3 sin B. Since the direction of the oscillation in the air space does not come into consideration, we are con- cerned only with the absolute value of this expression and not with its sign. The determination of the absolute values of the quantities a and £ would involve elaborate theoretical investigations, and is, moreover, unnecessary for the explanation of the experimental results.

Experiments with the Secondary Circuit in a Vertical Plane.— When the circle forming the secondary circuit was placed with its plane vertical, anywhere in the neighbourhood of the .primary, the following results were obtained.

442 DYNAMICAL THEORY OF INDUCTION.

The sparks disappeared for two positions of the air space, separated by ISOdeg., namely, those in which it lay in the horizontal plane through the primary ; but in every other position sparks of greater or less length were observed.

From this it followed that the value of a must have been constantly zero, and that 6 was zero when the air space was in the horizontal plane through the primary.

The electromagnetic lines of force must therefore have been perpendicular to this horizontal plane, and therefore consisted of circles with their centres on the primary ; while the electro- static lines of force must have been entirely in the horizontal plane, and therefore this system of lines of force consisted of curves lying in planes passing through the primary. Both of these results are in agreement with theory.