or for several bodies in contact, as well as for the case when the bodies are simply subject to their mutual influence. However, this determination has been made, in a rigorous and complete manner, only in the cases of an insulated sphere and an ellipsoïd, and in that of two spheres in contact; for the other cases, Poisson has merely indicated the general method, without having given its application. We must then take account of the electricity, developed by induction, which modifies the electric state of each body; but this modification, which occurs at the very instant when the bodies are placed in presence, does not prevent the electric state becoming permanent, so long as the latter do not change their place. It is further necessary, in order that this permanent state of equilibrium should remain, that the resultant of the actions exercised, upon any point taken in the interior, by the fluid strata which covers them, should be equal to 0. This condition furnishes, in each case, as many equations as there are conducting bodies under consideration; and these equations serve to determine the variable thickness of the electric stratum on the different points of these bodies.

It follows, from this mode of looking at the phenomena of static electricity, that the office of a conducting body consists simply in determining in the air, for the electric fluid, a vessel of which it is, so to speak, itself the mould; and the boundaries of which are the stratum of air that envelopes it. Free to travel in this vessel, the electric fluid distributes itself entirely towards the insulating sides, against which it is retained, exercising, however, a pressure from within outwards, against which the atmospheric pressure resists. This pressure depends at once on the number of particles included in the stratum; consequently, on its thickness, and on the repulsive force exercised by the particles of the exterior surface, a force which is itself proportional to this thickness. The pressure is therefore in a ratio composed of the thickness of the stratum and of the repulsive force of the surface, or proportional to the square of the thickness. It necessarily follows from this, that the pressure of the electric fluid against the air may be very different in certain points to what it is in others, according to the form of the body on which the thickness of

the stratum depends; and may even become infinite in one point in respect to that experienced by other points. It may hence happen that this pressure may surpass, in some parts of the body, the resistance opposed to it by the air; the air then gives way, and the fluid passes away, as it were, through an opening. This takes place at the extremity of points, and on the sharp edges of angular bodies; for it has been demonstrated that, at the summit of a cone, the pressure of the electric fluid would become infinite if the electricity could be accumulated there.

It is impossible for us to give here a more complete idea of M. Poisson's labours, which are entirely within the domain of pure mathematics. We shall confine ourselves to adding that Mr. Plana took up a portion of the subject treated of by M. Poisson, in a memoir on the distribution of electricity on the surface of two insulated conducting spheres, published at Turin in 1845. He examined the different cases that may be presented by the problem: that in which the spheres are in contact; that in which they are separated by any interval; that in which this interval by which they are separated is very small in comparison with the distance of their two centres. Mr. Plana has, in like manner, demonstrated in a more rigorous way certain admitted principles on the relations between the thickness of the electric curve, and the forces which emanate from it; as well as the theory of the proof-plane, against which objections had been made. In this great work, which is the fruit of a very deep mathematical analysis, the mass of the electric matter, and the form that it assumes, are the only elements taken into consideration. The calculation is completely independent of the cause, whatever it may be, which retains free electricity on the surface of conducting bodies. Mr. Plana, in a note which terminates his researches, shows that we may demonstrate à priori, by the simple general fact that free electricity distributes itself on the surface of conducting bodies, that the law of its repulsive force must be that of the inverse ratio of the square of the distance. This very elegant demonstration, which is based solely on a result of observation, replies victoriously to Sir Wm. Snow Harris's objections against the

law discovered by Coulomb; at least when it is circumscribed, as we have said, to the case of simple physical points.

Theoretic Explanation of the Movement of electrised Bodies; and the Electric Mill.

The repulsion and attraction of electrised bodies is a consequence of the repulsion and attraction exercised upon each other by the two electric fluids. In fact, if the electrised bodies are of an insulating nature, the fluids, not being able to separate themselves from the ponderable particles to which they are united, draw them with them in their attractive or repulsive movements; at least, when the electricity is not too feeble, or the bodies are not too heavy to obey these actions. But if the two bodies are conductors,-if, for example, they are pith-balls, we must refer to atmospheric pressure in order to explain their movements. Since the two balls have the same electricity, immediately they are brought near to each other their electricities repel each the other; and being able to travel freely, they arrange themselves into the most remote portions of the balls. There they each form a stratum which, acting exteriorly in a contrary direction to the atmospheric pressure, diminishes its effect; whilst this pressure experiences no resistance on the interior portions of the surface of each ball, from which the fluid has been driven. Being more pressed upon by the atmosphere from within outwards, than inwards from without, the two balls separate. When they have a contrary electricity, the two electricities convey themselves, by virtue of their mutual attraction, into the portions of the surface of each ball that are nearer to each other; and these diminish the effect of the atmospheric pressure, which, not being counterbalanced on the exterior part of the surfaces, drives the balls one towards the other. We can imagine that the same explanation is applicable to the case in which there is only one movable ball opposite to a fixed electrised body; it is equally applicable to the case in which the movable ball is not electrised; for, as we have already seen, it is first electrised by induction, and the action takes place between two bodies both electrised.

Finally, there is another case of movement in electrised bodies that we have not yet described, and which merits especial mention, on account of the interest it presents in a theoretical point of view. It is that which is realised by a small apparatus, known under the name of the electric mill.

The electric mill consists of a thin stem of metal, whose two ends are pointed, and bent in an opposite direction, and which is itself balanced upon a pivot by means of a cap fixed to its centre of gravity. The pivot is put into communication with the conductor of an electrical machine, and, as soon as the latter is in action, the stem acquires a rapid rotatory movement, as if the extremities of the points were powerfully repelled. In this experiment the electric fluid that is spread over the surface of the stems of the mill, exercises everywhere a pressure on the surrounding air; if it found no escape, the opposite pressures being always equal, the apparatus would remain at rest. But it escapes by the points, where it overcomes the atmospheric pressure; and, as it no longer exercises any pressure on the orifice of escape, the pressure, which continues to be exercised on the opposite point, causes the movement by a true recoil, analogous to that produced by liquids or gases upon reacting machines during their escape. We shall return to this explanation, for it is not altogether free from certain objections; and we shall see that the experiment of the Fig. 65. electric mill is probably connected with a more general phenomenon of dynamic electricity. Four or

Fig. 66.

five branches, instead of two, are given commonly to the electric mill (Fig. 65.); the form of the experiment may also be varied by making, for example, a stem that is terminated by two points turned in opposite directions, ascend along two wires by the ef

fect of the escape of electricity (Fig. 66.).

Examination of the Part attributed to atmospheric Pressure, in the Phenomena of static Electricity.

The part attributed by theory to atmospheric pressure in the phenomena of static electricity, is not free from all objection. Some facts that we are about to relate are but little favourable to it. M. Becquérel succeeded in developing by friction, and in maintaining under vacuum, a notable quantity of electricity in a gold-leaf electroscope. Although the air had been rarefied to th of an inch, the gold leaves remained diverging, and consequently retained free electricity, for more than two days. It is very difficult to admit that the small quantity of air remaining in the receiver had been able to exercise a sufficient pressure to produce this effect.

Hawksbee and Gray had also observed that electrised bodies attract light bodies, as well in vacuo as in air; but, as they were insulating bodies, we can prove from this nothing very positive against the action of atmospheric pressure, when conducting bodies are concerned. It is not the same with the experiments recently made by Sir W. Snow Harris. This philosopher had fixed a copper ball, of two inches in diameter, at the extremity of an insulated metal rod; and, after having placed it in the centre of a receiver, had put it in communication with an electroscope by another rod; he then gave to the ball a quantity of electricity such that the deviation of the electroscope was 40°. This divergence was maintained even when ths of the air had been removed from the receiver. The result was the same when the electroscope itself was placed under the receiver, and when it was electrised directly by means of an insulated metal rod communicating with the exterior. The divergence did not vary even when the air was reduced by 9ths of its original density. In both cases the gold leaves of the electroscopes gradually approached, when an insulated metal ball was brought near to the electrised body; but they separated again when the ball was withdrawn: this was evidently a simple effect of induction. In all these experiments the interior of the bell