shell. Supposing these to be of radii e and a, the applied pressures being p and p', we thus find

Since the dilatation of a layer is uniform throughout, it is

given by

A=dv/v.

We may thus write (68) in the alternative form

The relation we have established is of course insufficient by itself to determine the dilatation in any one layer of a compound shell or sphere, but it at least supplies a very simple check on the accuracy of results otherwise determined. For the three-layer shell (e.a.c.a1.b.a. a) it gives, denoting the dilatations in (e. a. c), (c.a1.b), and (b.a. a) by A, A1, A, respectively,

k▲(c3 —e3)+k ̧a1(b3 — c3)+k▲2(a3—b3)=pe3—p'a3,

and this will be found consistent with the values of the dilatation supplied by (38), (39), and (40) for the three layers.

Summary.

The principal results arrived at for the application of uniform surface-pressure are as follows:

A.

(1) In any simple sphere or spherical shell, or in any one layer of a compound sphere or shell, there is a simple linear relation, independent of the elastic constants of the material, between the values of any the same stress or strain over any three concentric spherical surfaces.

B.

For the effects of altering a spherical layer so as to increase one or both of its elastic constants (the effects of diminishing one or both constants, generally speaking, being the exact opposite) :

:

(2) The radial-pressure gradient falls off in steepness both nside and outside the altered layer, making up for this by ncreased steepness in the layer itself.

(3) The change of volume-whether expansion following nternal pressure, or contraction following external pressure -is reduced in the material both inside and outside the altered layer.

(4) The altered layer itself suffers greater change of volume than if the entire sphere or shell were composed of the same material as the layer.

(5) The alteration of compressibility only throughout a layer of given volume is equally effective for all positions of the layer, but the alteration of rigidity only is more effective the nearer the altered layer to the centre.

(6) The stress-difference vanishes at every point of a simple solid sphere; but, if the compressibility of a layer be altered, the stress-difference attains a finite value both outside the layer and in the material of the layer itself, becoming of importance in the layer itself however thin that may be.

XIX. Electrical Resonance and Electrical Interference.
By JOHN TROWBRIDGE*.

[Plate VII.]

HE solution of the differential equations which express

THE solution ofon of electricity on conductors of various

forms and varying magnetic permeability is not always simple; and many assumptions have been made in regard to the constants which enter these equations. Thus Poincaré† assumes that the spark of a vibrator is dampened more readily than that of the oscillator. This has been shown independently by Bjerknes to be true when the oscillating circuit is not closed by a spark-gap ‡. Stefan §, by making the assumption that the oscillations are confined to the outer layer of the conductor, reduced his differential equation to the form treated by Lord Kelvin, in periodic heat movements. Analysis leads him to suppose that the formula t=2π LC applies only to a special case; and he gives a more general law.

With the conviction, therefore, that the experimental side

*Communicated by the Author.

+ Electricité et Optique.

Ann. der Physik und Chemie, xliv. 1891; xlvii. 1892, $ Ibid. xli, 1890,

of the subject should be more developed in order to decide, if possible, upon the truth of the various assumptions that have been made, I have continued my studies on the oscillations of electricity with the aid of more powerful methods of studying periodic currents than I had hitherto used. In a paper on the oscillations of lightning discharges*, I expressed the opinion that the method first employed by Spottiswoode, of exciting a Ruhmkorff coil or transformer by means of an alternating-current dynamo, put in the hands of an experimenter a far more powerful method of studying electrical oscillations than the old method of charging Leyden jars by means of an electric machine, or by the use of a Ruhmkorff coil with a battery. I have, therefore, in this investigation, employed an alternating machine capable of giving 120 volts and a current of from 15 to 25 amperes, and have employed suitable transformers to obtain the necessary difference of potential to produce the sparks which I wished to study.

Generally I have employed one primary or exciting circuit between two entirely separate and disconnected resonating or secondary circuits. The image of the three sparks thus produced could then be compared upon the same plate.

Before entering into a more detailed account of the apparatus I employed, I will state the most striking results which I have obtained. A unidirectional spark (non-oscillatory) always excites an oscillatory discharge in a secondary circuit if the self-induction, capacity, and resistance of this secondary circuit permit an oscillatory movement. It is therefore not necessary that the spark in a primary circuit should be an oscillating one in order to excite oscillations in a neighbouring conductor. In this respect two electrical circuits are not in close analogy with two tuning-forks. It is difficult by a unidirectional movement of the prongs of one tuning-fork to excite the vibrations of another fork which is not in tune with the first fork. In every secondary circuit, or circuits neighbouring to the primary circuit, the first effect of the exciting unidirectional primary spark is to make the secondary circuits act as if there were no capacity in their circuits. In these circuits a thread-like spark results which is exactly like that produced when all the capacity in the secondary circuits is removed. After a short interval of time the electricity rushes into the condensers and begins to oscillate, the strength of the oscillations rising, after one or two vibrations, to a maximum and then decreasing; the rate of oscillation finally assumes a steady state, and is expressed by the formula t=2π √LC.

*Phil. Mag. October 1893.

This formula, moreover, does not hold for the first instants. The electricity seems to be separated only along the wires at first, and the circuit vibrates more like a closed organ-pipe than an open one.

If a unidirectional primary spark excites oscillations in neighbouring circuits which are slightly out of tune, the phenomenon of electrical beats or interferences can be produced in these circuits, and can be shown by photography.

If the primary spark ceases to be unidirectional and is allowed to oscillate, the oscillations of the primary spark tend to compel those of the secondary or neighbouring circuits to follow them; if they are not sufficiently powerful to do this, they beat with the oscillation of the secondary circuit. Moreover, if all capacity is removed from the neighbouring circuits, they oscillate in tune with the primary circuit, following the latter exactly. The secondary circuits without capacity act like sensitive plates and exactly reproduce every disturbance in the primary oscillating circuit. Leaving for the end of this paper a more detailed account of my results, I will now describe my apparatus.

Fig. 1.

BAC

D

T

M

In fig. 1, S, S', S" are three spark-gaps in the same vertical plane, but not immediately over each other, in order that the photographs may not overlap. B, A, C are three coils placed vertically on the same axis. These circular coils consisted of from one to four turns of well insulated wire. The mean radius of the coils was 915 centim. D, E, and F are the condensers respectively of the three independent circuits: these condensers were made of hard sheet-rubber 0.3 centim. thick; the coated surfaces could be varied by rolling up the layers of tinfoil which formed the coating. In certain cases aircondensers were substituted for the india-rubber condensers both for the condenser E of the primary circuit and for the condenser D of one of the secondary circuits. The air-condenser

employed for the primary circuit consisted of two large iron. frames, upon which sheets of tin were screwed. The condenser-plates thus made were 210 centim. by 330 centim., and were placed 2 centim. apart. The air-condenser employed in the secondary circuit replacing at times the hard-rubber condenser D is described, together with the photographic apparatus, in a previous paper on Electrical Oscillations*. It was placed in a room provided with yellow window-shades (orange fabric). It was found necessary also to construct a camerabox about 10 feet long, which extended from the spark-gap to the revolving mirror. This box was so constructed that the direct light of the sparks was shielded from the sensitive plate which was placed directly below the spark-gap. The plate thus received only the light thrown by the revolving mirror, which was placed at the opposite end of the camera-box. The operator, seated at the spark-end of this camera, closed for an instant a key K (woodcut, fig. 1) placed in the circuit of the alternator M. Looking through the film of the sensitive plate, one could determine when a suitable photograph had been taken; for the image of the sparks spread out by the revolving mirror could be clearly seen, the film acting like a ground-glass. It was thus possible to take a large number of photographs with the greatest ease, the rate of the alternator being so high that at each sudden make of the key several photographs could often be obtained at once upon the strips of sensitive plates, which were 25 centim. by 6 centim. It is evident that one of the circuits, for instance B, could be made a time-circuit. The self-induction and capacity in the circuit could be carefully determined and maintained constant. The formula t=2π √LC could be thus printed, so to speak, on each negative. For this time-circuit I have employed a fine wire coil which was slipped upon the same electromagnet which formed the primary of the step-up transformer T. I had thus two step-up transformers with a common primary one produced the sparks of the vibrator, the other the spark of the time-circuit.

In fig. 2 (Plate VII.) S' represents photographs of the unidirectional primary spark. S is the unidirectional spark produced in a neighbouring circuit B from which the capacity has been removed. S" is the oscillating spark in the circuit C: the condenser of this circuit was an air-condenser. The spark S shows that no oscillation is concealed by the heavy pilot-spark of the exciting spark S'. photographs S" show that the unidirectional spark S′ can *Proc. Amer. Acad. Arts and Sciences, May 28, 1890; Phil. Mag. [5] xxx. p. 323 (October 1890).

The