with a length of 1.8 meters. The quantitative results of these experiments are shown in Fig. 5, in which the abscissæ of the curve are the double lengths of the rectangles, and the ordinates represent the corresponding maximum sparking distances. The sparking distances could not be determined with great exactness, but the errors were not suffi cient to mask the general nature of the result.

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,

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

3

1000

FIG. 6.

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.

When the wire c d was surrounded by an iron tube, or 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.

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 a node at the center, 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 center of the circuit the sparking will be very slight, increasing as the sphere is moved farther 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 center of the circuit. The author then endeavored to produce a vibration with two nodes, and for this purpose

Fic. 7.

he modified the apparatus previously used by adding to the micrometer circuit a second rectangle e f g h exactly similar to the first (as shown. in Fig. 7), and joining the points of the circuit near the terminals by wires 13 and 24, 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 and 3, and 2 and 4, and a pair of nodes at the middle points of c d and gh. The vibrations 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 3 millimeters to about 1 millimeter. The existence of resonant action between the circuit C C' and the micrometer circuit was however fully demonstrated, for any alteration in the circuit e f 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 bƒ and the same sphere, showing that the nodes were in c d and g h, 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 belong 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 course not the fundamental vibration of the circuit, which con sisted 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 the 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 1 and 2 were broken off, leaving only the sparks between 2 and 4, which measured the intensity of the fundamental vibration. The period of vibration of the circuit CC' was then increased by drawing it out to its full length, and thereby increas ing 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 CC 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 corresponded 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 will in the same conductor either the fundamental vibration or its first overtone, to use the language of acoustics.

Hertz appears to consider it very doubtful whether it will be 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 themselves. 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 moreover 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 hammer, rather than in the comparatively slow harmonic vibration of tuningforks or strings; and in the case of vibrations 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, should other physicists desire to repeat the experiments, Dr. Hertz notes a fact of very considerable interest 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 invaribly diminished.

Theory of the experiments.-The theories of electrical oscillations which have been developed by Sir William Thomson, von Helmholtz, and Kirchhoff, have been shown* to hold good for the open-circuit oscillations of induction apparatus, as well as for the oscillatory Leyden-jar discharge; and although Dr. Hertz has not succeeded in obtaining definite quantitative results to compare with theory, 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; P its co-efficient of self-induction in absolute electro-magnetic measure expressed therefore in centimeters; C the capacity of one of its terminals in electro-static measure, and therefore also expressed in centimeters; and the velocity of light in centimeterseconds.

Lorentz, Wiedemann's Annalen, 1879, vol. vii, p. 161.

Then, if the resistance of the conductor is small,

In the case of the resonance experiments, the capacity C was approx imately the radius of the sphere forming the terminal, so that C=15 centimeters. The co-efficient of self-induction was that of a wire of length 1=150 centimeters, and diameter d= centimeter.

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

Putting k-1 this reduces to Neumann's formula; for k=0 it reduces to that of Maxwell; aud for k=-1 to Weber's. The greatest difference in the values of P 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 approximation it is enough to assume that k is not a large positive or negative number; for if the number 1902 does not give the correct value of the co-efficient for the wire 150 cm. in length, it will give the value corresponding to a conductor not differing greatly from it in length.

Taking P=1902cm., we have 7 √CP=531cm., which represents the distance traversed by light during the oscillation, or, according to Maxwell's theory, the length of an electro-magnetic æther wave. The value of T is then found to be (1,000) 1.77 hundred millionths of a second, which is of the same order as the observed results.

2 v

1.77

The ratio of damping is then considered. In order that oscillations may be possible the resistance of the open circuit must be less than /P. P. For 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 e2. fore be reduced in the ratio 1:2.71 in

rT

The amplitude will there

oscillations. Unfortunately we have no means of determining the resistance of the air space traversed by the spark, but as the resistance