Most of the phenomena which Electricity presents have been described at large in the Encyclopaedia; and the experimental part of the science remains nearly in the same state as at the publication of that work. This is not the case, however, with the theoretical. The theory was then founded on suppositions more or less doubtful; on ingenious but contracted views of the subject; and rather on empirical relations among the phenomena than on calculations rigorously mathematical. It is to supply this defect, that we must especially devote ourselves in this Supplement, and happily the progress of science affords ample materials.

To proceed methodically in the concentration of the electrical phenomena, let us recol successively all the general laws in which they are recorded, and which are established by experiment. The first consists in the mutual attraction or repulsion which electrified bodies exert, or seem to exert, towards each other. These properties are exhibited when we electrify a tube of glass or of resin, by rubbing it on a woollen cloth, and then touch with this tube small and light balls of wood, or the pith of the elder, suspended in a dry state of the air by threads of silk equally well dried. In this case, the silk prevents the escape of the electricity, and the air also possesses the same faculty. The little balls being only in contact with insulating bodies, are thus protected from any waste of their power. They are what we term insulated. But the moment the electrical property is communicated to them, they mutually fly asunder, and, contrary to the tendency of gravity, recede from the vertical; precisely as if the electricity which attaches itself to their surfaces had determined them to repel each other.

The result is alike, whether the little balls have been touched with the tube of glass or with that of resin, provided both are touched with the same. But, if one of them be touched with the tube of resin, and the other with the tube of glass, these two tubes having been both rubbed on a rubber of the same nature, then the two balls approach each other, contrary to the tendency of gravity, as if by a mutual attraction. This result being opposite to the first, obliges us to distinguish two modifications of electricity different from each other, at least in the apparent effects which they produce. These are what have been termed positive and negative electricity. We shall not employ these terms on account of their already presenting to the mind the ideas of addition and subtraction, which are really hypothetical, since they go beyond the facts. To express the distinction between the two kinds of electricity, we shall name them according to the method which serves to develop their influence. We shall call that vitreous electricity which a tube of glass exhibits when rubbed on a rubber of wool, and that resinous electricity which is obtained on rubbing upon the same rubber a tube of resin; though either of these electricities could be yet produced by some other proceeding different from what we have indicated in this definition.

We must now seek, by experiment, for the law according to which these attractions and repulsions operate, that is to say, what their relative intensity may be at different distances. The separation of the little balls furnishes an index of this; but it is not sufficiently delicate, and, as we shall soon see, too complicated, to measure it with exactness. The same may be said of all the contrivances in which weights so small as they must necessarily be, are made to balance the attraction or the repulsion. The torsion of metallic wires is the only force sufficiently minute, sufficiently constant, and whose effect can be estimated with sufficient ease, to be employed with advantage in measurements of this kind. Such is the object of the electrical balance of Coulomb, described in the Encyclopedia under the word Electrometer, and represented in the annexed figure. (1.) In what follows, we shall suppose the reader to have that description before him.

A small horizontal lever ab, formed by a very fine thread of gum lac, a powerful insulating substance, is suspended by its centre to a metallic wire, CS, two or three feet long. This lever at one of its extremities, a, carries a little ball of the pith of the elder, or a small plane of gilt paper, balanced by a slight swelling formed at b, the extremity of the other arm. Electricity. The small lever being supposed thus freely suspended and balanced, if we attempt to turn it any number of degrees in the horizontal plane in which it is situated, this cannot be done without twisting in the same degree the metallic wire SC, to which it is attached. But this wire, by virtue of its elastic reaction, will resist the torsion with a force, depending on its dimensions, on its nature, and on its physical constitution; and, according to Coulomb's experiments, so long as its state of aggregation has not received a permanent alteration, by twisting, its resistance will be exactly proportional to the angle by which we withdraw it from its natural position of rest; proportional, therefore, in this case, to the angle which we cause the small horizontal lever ab, to describe. (See Biot, Traité de Physique Experimentale, Tom. I. and II.) We have only, therefore, to produce this angular deviation by means of the electrical repulsion, and the torsion of the wire will measure its effect. For this purpose, the lever ab, being in that position of rest to which the natural equilibrium of the wire tends to carry it, we place beside the ball a, another ball A, (fig. 2.) of the same or of a different diameter, but fixed upon an immovable and insulating support, formed also of a short and very fine thread of gum lac. The two balls being now in contact, we touch the fixed ball A, with the extremity of an electrified tube. The electricity then divides itself between the two balls, the one fixed, the other moveable; these fly from each other, and this repulsion forces the lever, ab, to turn round its centre, (fig. 3.) until the repulsive force, weakened by the distance, is exactly balanced by the force of torsion of the suspending wire SC. Hence arises a state of equilibrium which is attained after several oscillations. We now observe, upon a divided circle which surrounds the apparatus, the arc Aa, which the lever has described; and turning the button S, by which the wire is suspended, and which is itself divided on the circumference, we twist it round a certain number of degrees until the angular distance of the balls becomes the half, the fourth, the eighth, or any other fraction of what it was at first; and then by comparing the degrees of torsion with the angles where the ball a stops in every case, we obtain the relative values of the repulsive force at various distances. In this manner it has been found by Coulomb that this force, like that of the celestial attraction, is rigorously proportional to the quantity of free electricity upon each body, and reciprocally as the square of their mutual distance. An experiment of a similar kind demonstrates that the same law extends also to the attractive force of the bodies charged with different electricities; and the sensibility of the apparatus is such, that no doubt can remain of the accuracy of the result. The only source of error would arise from the waste of electricity which is continually going on, through the air, and through the supports of gum lac, which, though highly insulating, can never be obtained in this respect altogether perfect. But the above mode of experimenting may itself serve to estimate the effects of these two causes. For we have only to leave the two bodies a and A near each other for a considerable time, several hours for example, taking care to untwist, from time to time, the wire SC, in order to diminish its resistance, and replace the balls at the same distance from each other in proportion as their repulsive force diminishes by the progressive waste of the electricity which covers them. It is easy to conceive that experiments of this nature, properly varied, ought to show the law of the waste of electricity through the air and along the supports. It was, accordingly, in this manner that Coulomb determined it; he was thus enabled to correct all his results, and to bring them up to the mathematical case of air perfectly dry, and of supports completely insulating; and it is with these corrections that the law of the squares of the distances comes out with the utmost rigour, as above enunciated. The apparatus employed by him for this purpose, and which we have explained (fig. 2, 3), is called the electrical balance.

What renders the torsion of the metallic wires peculiarly adapted for these kind of experiments is its extreme sensibility, by which the slightest variation in the intensity of the attractive or repulsive force produces an enormous change in the angle of torsion, which we must communicate to the metallic wire in order to balance it. In the experiments, for example, related by Coulomb,—to turn the lever ab a whole circumference; it was only necessary to apply to one of its extremities a force equal to the weight of \( \frac{1}{79} \) of a Troy grain.

Another remarkable result presents itself in these experiments. Whatever be the matter of which the balls are composed, provided that they are electrified simultaneously as we have supposed, the electricity divides itself between them always in the same ratio. If they are equal in volume, this ratio is that of equality; if they are unequal, the allotment follows a proportion depending on their diameters; but the chemical nature of the substance has no influence whatever. This may be proved by the following experiment: When the moveable ball a, of the balance, has been electrified simultaneously with the fixed ball A, and has receded to a certain distance, where the repulsive force maintains it in equilibrium with the force of torsion of the wire; touch the fixed ball A, with another ball of the same diameter insulated at the extremity of a small cylinder of gum lac, and withdraw it immediately after the contact; the electricity of the fixed ball will now be reduced exactly one-half, as the moveable ball will indicate by the new distance at which it will place itself. Now, this same reduction of one-half takes place whatever be the chemical nature of the ball with which the contact is effected. This fact proves, that the electricity is not retained in the balls by a chemical affinity for the material substance of which they are composed; and thus furnishes one great character which the theory must admit and represent.

In place of balls, we may equally well employ in these experiments small circular disks of gilt paper; or any other bodies, whatever be their form, provided their dimensions are very small, compared with the distance at which we make them act upon each other: this condition is, in fact, the only one which is necessary for making all the quantities of electricity which they possess to act together as if they were united in one single point. But the spherical form, and the circular, have peculiar advantages, arising even from the very manner in which electricity disposes itself in the bodies where it is in equilibrio, as we shall afterwards explain.

This mode of disposition is the first thing to which we must now attend. For, if we succeed in determining it by experiment, it will furnish us with geometrical characters, which the theory ought to satisfy, and which consequently will throw light upon the conceptions by which electricity may be represented. This determination is still obtained by means of the torsion balance. For this purpose we prepare a small circular disk of gilt paper, P (fig. 4.) which we insulate by fixing it to the extremity of a very fine cylinder of gum lac, SP; we insulate also, as completely as possible, the body which we mean to study, and then electrify it by communicating a spark drawn from a Leyden phial, or an electrophorus. If we now wish to know the state of any of the points of its surface, we touch this point with the little gilt disk, which we shall call the proof plane, and which, in this experiment, we hold by the extremity S of its insulating support, SP. We then carry this plane in the electrical balance, where the moveable ball has been previously charged with an electricity of the same kind, and place it for an instant in contact with the fixed ball. We then withdraw it, and the fixed ball being now electrified in the same manner as the moveable one, repels this with a force measured by the angle of torsion, at which the moveable ball stops. While the little plane and the balls of the balance remain the same, the division of the electricity between the little plane and the moveable ball, follows a constant proportion; thus the repulsive force which results, and which drives off the moveable ball, is proportional to the quantity of electricity with which the little plane is charged. But, moreover, experience proves that this is proportional to the quantity of electricity which really exists at the point of the body where we made the contact. The repulsive force, therefore, exerted by the fixed ball, is, likewise, proportional to that quantity of electricity, and thus the torsion observed will give us its measure. By repeating the same proof upon various other points of the body which we submit to experiment, we may determine the manner in which the electricity is distributed throughout.

This supposes only that the body and the balls of the balance are perfectly insulated, and lose none of their electricity while the experiments last. This constancy can never be made to hold with rigour, but we can correct its effect in a very simple manner. For this purpose, compare all the points m, m', m'', of the body with a single point, which we shall denote by M. To compare the state of m, begin by touching M with the proof plane, and determine by means of the torsion balance, the intensity of the repulsive force. Then, having deprived the fixed ball of the electricity which had been communicated to it in this experiment, touch m with the proof plane, and determine, in like manner, the repulsive force which results. Observe the time that elapses between these two comparisons, and suppose, for example, that it is a minute. In a minute after the second experiment, try again the state of M, and take the mean of the two values of the repulsive force, which this point will have furnished. The second will be weaker than if M and m had been both tried at the same time; but the first will be stronger; and if the waste is very slow, as the precautions we have indicated suppose it to be, the arithmetical mean between the two results will be the same that would have been obtained at the intermediate instant, that is to say, when the observation was made upon the point m. This corrective process, so simple and so exact, was imagined and employed by Coulomb, who, in general, has left but little to be added as to the use of his ingenious apparatus.

The method of alternate contacts which we have explained, may be employed to discover the disposition of electricity, not only at the surface of bodies, but even in their interior. For this purpose, it is sufficient to pierce in the body a small canal, terminating on its surface, and to plunge the proof plane to the bottom of this canal, when the body is electrified and insulated. In this manner the remarkable result has been obtained, that, whatever be the external figure and substance of the body, provided it be a conductor, the electricity will remain nowhere in its interior. It will confine itself entirely to its surface, where it will form a stratum infinitely thin; and this fact, demonstrated by Coulomb, forms another capital basis for the theory.

In regard to the distribution on the surface, it depends on the form of the body. In spheres, for example, the electricity is distributed in a spherical stratum of a constant thickness. On an ellipsoid of revolution, the thickness is variable. The exterior surface of the stratum is that of the body itself, and the quantities of electricity at the extremities of the greater axis are proportional to its length. Lengthen, then, considerably, the ellipsoid, and the thickness of the stratum will augment at the extremities of its greater axis in the same proportion. If the resulting force of repulsion is then sufficiently powerful to overcome the resistance which the ambient air presents to it, the electricity will escape by the points; and this gives the reason that it escapes at the corners of all angular bodies. In general, whatever be the exterior form of the electrified body, this form constitutes the exterior surface of the electrical stratum. As to the interior surface, it always differs infinitely little from the first. But its deter- mination can only be expressed by analytical conditions, deduced from theory, and which experiment demonstrates to be conformable to the truth.

The method of alternate contacts serves also to measure another phenomenon of capital importance. This is the development of electricity in conducting bodies by the influence at a distance, of bodies already electrified. When a body A, for example, which we shall suppose, for greater simplicity, to be spherical, has been electrified and insulated, if we then bring within a certain distance of it a body B, a conductor of electricity, and equally well insulated, but not previously electrified; this latter, as it approaches the other, begins to give signs of electricity; it ceases, however, to do so when we remove it again to a great distance, and it recovers this virtue, or loses it, according as we expose it to, or withdraw it from, the influence of the body already electrified. The philosophers who first discovered this fact supposed, in order to explain it, that every electrified body sent forth electrical emanations, which spread around it like an atmosphere, and electrified every conducting body which one might plunge into it. But as this effect is produced even through bodies such as glass, for example, which do not allow the electricity to pass through their substance, it is evident, that, if we wish not to go beyond the facts, we ought not to view this but as a certain influence exerted at a distance, like what is observed in the celestial attraction, and in the magnetic attractions and repulsions. Now, to study the effects and the laws of this influence, we must place the conducting and insulated body B, before the electrified body A, and so near it as to give evident signs of electricity; then touch different points of its surface with the proof plane, and study with the electric balance the nature and the quantity of the electricity which is developed in each of them. In making this experiment we find the following result: The electricity developed in B is not every where of the same nature; it is similar to that of A in the portion of B which is most distant from A, and different in that which is nearest. If A, for example, is charged with vitreous electricity, the anterior part of B, that towards A, is in the resinous, the posterior, in the vitreous state; and between these two extremes each of the two electricities extends in a certain zone, which may be determined by the proof plane. But in all cases the total quantities of the different electricities distributed over these zones have the remarkable property of being exactly equal, so that, if left to themselves, they would neutralize each other's effect. This supposes that B, before being submitted to the influence of A, was in the natural state; but if it has already received a certain quantity of electricity, we then find that the sums of the vitreous and resinous electricities which it possesses under the influence of A, do not differ from each other but by this primitive quantity; and hence we re-obtain it entire when we withdraw B from the influence of A, and deliver it to its natural state of electrical equilibrium. In every case, however, as it is natural to expect, this constancy only holds when we suppose the body B to lose none of its primitive or acquired electricity, either by the contact of the air, or the imperfection of the insulating supports. It is understood, therefore, that we correct the effect of these two sources of waste, by the laws already mentioned, which are determined for each of them by experiment. With the same condition we may study the electricity of A by the proof plane, either when it is abandoned to itself, or when it acts on B; and in this manner, we find that this action takes away from it absolutely nothing of its primitive electricity; but that when B is near A, and influenced by it, A is also influenced in its turn, so that the electricity we have given it is distributed in a different manner; and if the distance of B is small, or the reciprocal action energetic, the electricity ceases even to be of the same nature over all the surface of A. The part nearest to B takes the contrary electricity to that which the anterior part of B possesses, and the more distant takes the opposite. In a word, everything here is reciprocal, in regard to the two bodies, and the effects are only in proportion to the difference of their form, and of the quantities of external electricity which they primitively possessed.

The enunciation alone of these results is sufficient to point out the consequences. 1st, Since the body B takes nothing from the electrified body A, it must possess within itself the principles of the two electricities which are developed in it by the influence of this body. 2dly, Since these two electricities disappear when the influence of the external body ceases, although they cannot escape into the ground, on account of the insulation of B, their proportions must be such, that, being left to themselves, they will mutually neutralize each other. 3dly, This neutralization must operate without destroying these electricities, since they appear anew and quite entire every time we submit the body B reduced to its natural state, to the influence of the electrified body A.

We are thus led to discover, that the principles of the two electricities exist naturally in every conducting body, in a state of combination which neutralizes their effects. This we shall henceforth call the natural state of bodies. We see also that friction, which seems, at first view, a method of generating the two electricities, serves only to disengage them from their mutual combination, and to render the one of them sensible by absorbing the other; and this is the reason, without doubt, that we constantly observe the rubbing and the rubbed body exhibit contrary electricities. In fine, since the sole influence of an electrified body presented at a distance to another body in the natural state, forces the two electricities of that body to separate and to distribute themselves in such a manner that those of a different nature become the nearest to each other upon the two bodies, and those of the same nature the farthest, we must, in order to enunciate this fact, admit, that the electrical principles of a different name attract, and those of the same name repel, each other, according to laws which experience may perhaps enable us hereafter to determine.

These observations lead to another important consequence. When we begin to examine the electrical phenomena, we perceive that electrified bodies Electricity attract, or seem to attract, all the light bodies which are presented to them, without there being any necessity, for this purpose, of developing first the electric faculty in these, either by friction or communication. But we must now conceive, that this development goes on of itself, by the sole influence at a distance of the electrified body, on the combined electricities of the little bodies which we present to it; so that, even in this case, the attraction, whether real or apparent, that we observe, does not take place but between bodies that are really electrified.

Moreover, the developement of the combined electricities in these circumstances is indispensable, in order that the attraction may take place; for the latter is always the less vigorous as the former is effected with less facility. To be convinced of this, take two threads of raw silk, very fine, and of equal lengths. Suspend to them two little balls of equal dimensions, of which the one is of pure gum lac, and the other also of gum lac, but gilt on its surface, or coated with tin-foil; these two pendulums being then placed beside each other, and at a small distance, bring near them a tube of glass or of sealing wax, rubbed and electrified; you will see that the ball covered with metal, and on whose surface the decomposition of the combined electricities goes on with facility, will be much more readily and more vigorously attracted than the other. This one will not begin to be attracted till after a certain time, when the decomposition is finally effected on its surface in spite of the resistance which its substance opposes to the motion of electricity; and, in like manner, when once the decomposition has taken place on this ball, its electrical state subsists even after we withdraw the electrified body. The first ball, although gilt, contracts also in this manner a permanent electricity, because the resin of which we suppose it composed, impregnates itself with the electricity developed at its surface; and both of them are favoured in this respect by the contact of the air, which, under the influence of the electrified tube, tends, above all, to take away from them that part of their combined electricities which is repelled by this body; while it has less effect on that of which the repulsive force is disguised by the tube's attraction. Hence we observe, in general, that insulated bodies which have been for some time under the influence of an electrified body come, at last, to have an excess of electricity of the opposite kind, and of which the effects become manifest when we again withdraw them from the influence of this body.

As the results to which we have now arrived are of continual use in the developement and connection of electrical phenomena, it is necessary to reduce them to a kind of theorem, which we shall enunciate in the following manner:

When a conducting and insulated body B, in its natural state, is placed near another body A, electrified and insulated, the electricity distributed over the surface of A acts, by influence, on the two combined electricities of B, decomposes a quantity proportional to the intensity of its action, and resolves it into its two constituent principles. Of the two electricities thus liberated, it repels that of the same and attracts that of a different name with itself. The first diffuses itself over that part of the surface of B which is farthest from A, and the second over that which is nearest. Both being now at liberty, they act in their turn on the free, and even on the combined electricities of A, which, by this re-action, are partly decomposed, and, if A is also a conductor, are separated. This new separation produces a new decomposition of the combined electricity of B, and the same process goes on until the quantities of each principle, liberated upon the two bodies, attain an equilibrium, by a balance of all the attractive and repulsive forces which they exert on one another, in consequence of the similar or contrary nature of each.

Having thus discovered, in general, the attractive and repulsive properties belonging to the two electricities, the vitreous and the resinous,—having discovered their natural state of combination in bodies,—their separation by the influence of a body at a distance, and the general consequences which result from these new properties, we must, according to the philosophical method now adopted in the sciences, endeavour to submit them to calculation; so as to ascertain exactly the detail of the facts, and to anticipate, for example, in regard to each of the electrified bodies which we cause mutually to act on one another, what will be, on any point of its surface, the quantity and the nature of the electricity.

But as we have found that the effects of these mutual influences, such as we have observed them, arise from actions between the electrical principles themselves, it is easy to conceive that we cannot arrive at their cause without determining the nature and the mode of action of these principles; or, what is the same thing to us, imagining, according to the phenomena observed, some calculable mode of action which will represent exactly the phenomena, and which can be verified, if not immediately as to its physical existence, at least indirectly, but with certainty, in its consequences.

But if we consider the extreme facility with which the two electricities, the vitreous and the resinous, diffuse themselves in conducting bodies, and advance towards their surfaces where they are retained by the pressure of the air; if we consider the perfect mobility with which these two principles approach and recede from each other, unite themselves, or separate, without losing any of their original properties; it will be obvious, that the most probable idea we can obtain of their nature, is to regard them as fluids of perfect fluidity, whose particles are endowed with attractive and repulsive powers, and which arrange themselves in the bodies where they can move with liberty, in such a manner as to be in equilibrio, by virtue of all the interior and exterior forces which act upon them.

It is easy to see that each of these fluids must possess in itself a cause of repulsion, which tends to separate its particles from each other. For, if we suppose a certain quantity of vitreous or of resinous electricity, introduced into a sphere of metal where its motions are free, we know that it will diffuse itself entirely at the surface, where it will form a stratum of a very small thickness. If we augment the diameter of the sphere, the electrical stratum will always recede more and more from its centre, diminishing at the same time its thickness. If we at last withdraw altogether the pressure of the air, the electricity will be completely dissipated. These effects indicate with certainty a repulsive action exerted between the electrical particles of the same nature, and all the phenomena in which the combined electricities are separated from each other by influence at a distance, confirm completely this result, and at the same time also demonstrate the existence of a reciprocal attraction between the electricities of a different nature.

We see also from these phenomena that the attractions and repulsions become weaker in proportion as the distance increases; but according to what law? Among all those which can be tried, there is one which represents and reproduces completely all the phenomena; it is that of the inverse ratio of the squares of the distances. Adopting this law then, the constitutions of the two electrical principles will be comprised in the following enunciation: Each of the two electrical principles is a fluid, whose particles, perfectly moveable, mutually repel each other, and attract those of the other principle, with forces reciprocally as the square of the distance. Also at equal distances the attractive power is equal to the repulsive. This equality is necessary in order that in a body, in the natural state, the two combined electricities may not exert any action at a distance.

Having thus defined very precisely the characters and the mode of action of the two fluids, we must now explain the mathematical consequences of this definition, in order to compare them with the phenomena, and to see if they are exactly conformable to them. We must endeavour, above all, to find those which, being susceptible of a precise and numerical value, admit of greater rigour in their verification. But these deductions cannot be obtained but by very profound calculations, which require all the resources of analysis; and, even with the aid of these, it is only of late that they have been established in a general and exact manner. It is to M. Poisson that this fine discovery is due. We shall take from his treatise, published in the Memoirs of the Institute of France for 1811, the precise results which calculation has made known to him; we shall borrow them as the rigorous deductions of our first definitions; and it will then only remain for us to ascertain if they agree with the facts.

We shall begin with considering a single conducting and insulated body charged with an excess of vitreous or of resinous electricity, and exempt from all external influence. Setting out from the constitution assigned to the two fluids, calculation informs us, that the fluid introduced into this body will diffuse itself entirely on its surface, and will there form a stratum extremely thin. This is confirmed by observations the most minutely exact. Calculation determines also the interior surface of this stratum and its thickness. The exterior surface, bounded by the air, is the same with that of the body. The air is in this case to electricity, as an impenetrable vase of a given form, which contains it in its interior capacity, and resists by its pressure the tendency which it has to escape.

The interior surface is in every case but very little different from the other, the electrical stratum being very thin. But in order that the electrical state of the body may remain permanent, the form of this surface must be such, that the entire stratum exert neither attraction nor repulsion on the points comprised within its cavity. For, if these actions were not reduced to nothing, they would operate upon the combined electricities of the body, would decompose part of them, and the electrical state of the body would therefore change, contrary to the state of permanency which we have supposed. The analytical condition which establishes this property, determines the form and the thickness of the stratum, which may, and, even in general must, be unequal upon the different parts of the surface of the electrified body. (See the Memoirs of the Institute of France for 1811.) If the body, for example, has the form of a sphere, the two surfaces of the electrical stratum will be spherical, and will have their centre in the centre of the sphere. The thickness of the stratum then will be everywhere constant, and equal to the difference of their radii. Newton, indeed, has long since demonstrated, that, in the law of the square of the distance, such a stratum exerts no action on the points which are within. (Princip. Math. Lib. I. Prop. LXX.)

If the proposed spheroid is an ellipsoid, the interior surface of the electrical stratum will be also an ellipsoid, concentric and similar; for it can be demonstrated, that an elliptical stratum, of which the surfaces are also concentric and similar, exerts no action on a point situated in its interior. The thickness of the stratum in every point is determined generally by this construction. It hence follows, that this thickness is greatest at the extremity of the greater axis, and least at the extremity of the smaller; and the thicknesses corresponding to the two extremities of the different axes, are to each other as the lengths of these axes, which, as we have seen, is conform to the experiments. In general, the exterior surface of the fluid stratum is given by the surface of the body itself; and the whole problem is reduced to find for the interior surface a form very little different from this, which shall bring to nothing the total action of the stratum on all the points comprised within its cavity.

The electrical stratum thus disposed, acts by attraction and repulsion on the other electrical particles situated beyond its exterior surface, or at this surface itself. It attracts them if they are of a different nature from its own, and if they are of the same nature it repels them. This last case is that of the electrical particles which form the exterior surface of the stratum, each of these being repelled from within outwards, with a force proportional to the thickness of the stratum at that point. The particles situated under the surface, in the thickness of the stratum itself, suffer a similar repulsion, but weaker, as it is only proportional to the thickness which separates them from the interior surface of the stratum, for the particles with which they are enveloped on the side of the exterior surface, according Electricity- to the form of the two strata, exert on them no action at all. All these repulsive forces gradually decreasing, and being resisted in their effects by the external air, which opposes the escape of the electrical particles, it is easy to conceive, that there must result a total pressure exerted against this air, and tending to drive it off. This pressure is in a ratio compounded of the repulsive force exerted at the surface, and of the thickness of the stratum; or, as the one of these elements is always proportional to the other, we may say that, in every point, the pressure is proportional to the square of the thickness. It may therefore in general be variable on the surface of electrified bodies.

If this pressure is everywhere less than the resistance which the air opposes, the electricity is retained in the vase of air, and cannot escape. But if the pressure, in certain points of the surface, comes to exceed the resistance of the air, then the vase breaks, and the fluid escapes through the opening. This is what happens towards the extremities of the points, and on the sharp corners of angular bodies. For it can be demonstrated, that at the summit of a cone, for example, the pressure of the electric fluid would become infinite, if the electricity were allowed to accumulate there. At the surface of an elongated ellipsoid of revolution, the pressure does not become infinite at any point; but it will be so much the more considerable at the two poles, as the axis which joins them is greater in relation to the diameter of the equator. According to the theorems already cited, this pressure will be to that which takes place at the equator of the same body, as the square of the axis of the poles is to the square of the axis of the equator; so that, if the ellipsoid is very much elongated, the electric pressure may be very feeble at the equator, while at the poles it will surpass the resistance of the air. Hence, also, when we electrify a metallic bar, which has the form of a very long ellipsoid, the electric fluid runs principally towards its two extremities, and escapes by these points, in consequence of its excess of pressure above the resistance of the air which opposes it. In general, the indefinite increase of the electric pressure in certain points of electrified bodies, furnishes a natural and exact explanation of the faculty which points possess, of dissipating with rapidity into the non-conducting air the electric fluid with which they are charged.

If the nature of the electrified body were such that the electricity could not move freely in it, the excess of pressure, of which we have been speaking, would exert itself against the particles themselves of the body which envelope the electric stratum; or, in general, against those which, either by their affinity, or by any other mode of resistance, would oppose its dissipation.

Having determined, according to the theory, the manner in which electricity disposes itself in a single conducting body, insulated and unaffected by any external influence, let us pass to the more complicated case, where several electrified and conducting bodies act mutually on each other; and as it is necessary to make choice of bodies whose form renders the phenomena accessible to calculation, let us consider two spheres of some conducting substance, both electrified and placed in presence of each other at any distance.

The disposition of electricity in these circumstances, and in all those where several electrified bodies are submitted to their mutual influence, depends on a general principle, evident in itself, and which has the valuable advantage of reducing all these questions to a mathematical condition. The following is its enunciation, which we take from the treatise of M. Poisson.

If several electrified bodies are placed near each other, and if they arrive at a permanent state of electricity, it is necessary in this state that the resulting effect of the actions of the electric strata which cover them, upon any point taken in the interior of these bodies, be nothing. For if this resulting force were not nothing, the combined electricity which exists at the point in question would be decomposed, and the electrical state would change, contrary to the supposition which we have made of its permanency.

This principle, translated into the language of the calculus, furnishes immediately as many equations as we consider bodies, and as there are unknown quantities in the problem. But their resolution often surpasses the powers of analysis. M. Poisson, however, who has so happily discovered the general key of this theory, has at last surmounted all the analytical difficulties, for the case of two spheres placed in contact or near to each other, and primitively charged with any quantities of electricity. The formulae to which he has arrived afford a great number of results which can be verified by experiment, and which form so many severe tests of the justness of the theory. Besides the interest which such verification must always present, we will obtain in them the farther advantage, of fixing our ideas with precision on the most delicate phenomena which electricity produces.

Suppose, first, the two spheres in contact, and charged with either electricity, vitreous or resinous; calculation shows that there is no free electricity at the point of contact. From thence the thickness of the electric stratum goes on increasing on each of the spheres, according to a law which depends on the relation of their radii, but it attains always its maximum on the opposite side, on the line of the two centres. If we separate the two spheres, each of them preserves the same quantity of electricity which it has attained during the contact, and these quantities have to each other a relation which the calculation assigns according to the proportion of the radii.

The verification of these results is effected with the greatest facility, by means of the small proof plane, and by the general method of alternate contacts explained above. In this manner the indications of the theory are found to be confirmed by experiment in their minutest details; that is to say, that, on introducing into the theoretical formulae the diameters of the spheres on which we operate, or only the relations of these diameters, the calculus shows in advance, and as exactly even as the observations themselves, the law of the distribution of Electricity. electricity over the two spheres, as well as the proportion of its intensity on each. There is no occasion, even for this purpose, of new experiments; for we may take those which have been already made by Coulomb, and have for a long time been published in the Transactions of the Academy of Sciences; and, accordingly, it is with these previous results, whose priority, indeed, gives them all the authenticity of an incontestible fact, that M. Poisson has compared the numbers given by his theory.

The case of contact being thus completely analysed, let us examine what takes place when the two spheres separated from their contact are removed to a certain distance from each other.

In this case, a very remarkable phenomenon, discovered by Coulomb, is developed, and of which that philosopher determined the measure by his electric balance. We have seen that, during the contact, the electricity is the same on both the spheres. To fix our ideas, suppose it to be the vitreous. As yet its intensity is nothing, as we have seen, at the point of contact; but the instant we separate the two spheres, if their dimensions are unequal, this state of things ceases. A part of the combined electricities of the small sphere is decomposed, and what is of a contrary nature to that of the large sphere, namely, the resinous in our example, appears in, and near the point where the contact took place. This effect diminishes in proportion as we remove the spheres from each other, and vanishes altogether at a certain distance, which depends on the proportion of their radii. At that distance, the point of the small sphere, where the contact was made, returns to the state in which it was during the contact itself, that is, it has no more any species of electricity. From this position, if we still increase the distance, the electricity remains of the same nature over all the extent of the small sphere, and of the same nature also as it was during the contact. These changes are always peculiar to the smallest of the two spheres, whatever be the quantity of electricity which has been previously communicated to either. As to the larger of the two spheres, the electricity is always, and everywhere, of the same nature as at the moment of contact.

To observe these phenomena, we place the two spheres upon solid supports of some insulating substance, and of such a magnitude that their centres may be in a horizontal plane, as represented in the figure.(5.)

After having placed them in contact, and electrified them together, we remove them to a small distance, A a, from each other. We then carry the proof plane first to A upon the large sphere, to the point where the contact was made; and by trying, not with the electrical balance, but with a very sensible electroscope, the electricity which it carries off, we observe that it is always of the same nature with that upon the rest of the surface ABDE. Making then the same trial at the corresponding point a, we find that the electricity in this point is, first, of a nature contrary to that of the large sphere; then nothing when the distance becomes a little greater; then of the same nature when the distance becomes still greater; and in this manner, by gradually augmenting the distance, we see the phenomena pass through all the periods which have been already pointed out by the theory. These trials must be all made with the most sensible electroscope, in order to perceive the feeblest signs of either electricity.

In one of Coulomb's experiments, the large globe was twelve inches in diameter, and the small one eight. As long as the distance A a was less than an inch, the point a gave signs of a contrary electricity to that of the large globe. When the distance became equal to an inch, the electricity of this point became equal to nothing, as at the instant of contact; and, lastly, at every distance beyond this, it became of the same nature with that of the other sphere. The large globe remaining the same as in the preceding experiment, Coulomb gave to the small globe a diameter of only four inches; then the two electricities continued of an opposite nature to the distance of two inches. When this diameter was only two inches and under, the opposition was kept up to the distance of two inches five lines, but no farther.

The comparison of these experiments shows, that the distance A a, where the opposition of the electricity ceases, diminishes in proportion as the two globes approach to an equality, and becomes nothing when they become equal. This is a circumstance which is also confirmed by the theory, which equally indicates all the other details of the phenomenon, the relation between the radii of the two globes being given.

That these experiments may succeed well, the air must be very dry; else the electricity of the large globe, escaping through the air, will tend directly to neutralize the weak electricity of the opposite kind, which is developed at the point a of the little globe, and the phenomena will become much less sensible, if not entirely disguised.

We have hitherto supposed the two globes to have been brought into contact before being submitted to their mutual influence. This condition establishes between the quantities of electricity, which they possess, a relation that limits the generality of which the problem is susceptible, and to embrace it entirely, we must consider two spheres charged, in any proportions whatever, with electricity of the same or of a different nature; and that they gradually approach each other until they come into contact.

Here the analysis of M. Poisson has anticipated the results of experiment. It is from thence that we shall draw the details into which we are to enter; and which, if they should come one day to be observed, will furnish the severest test to which this analysis can be submitted.

When two electrified spheres are made gradually to approach each other, and when there does not exist between the species and the quantities of electricity which they possess, the particular relation which would be established by their contact, the Electricity. thickness of the electric stratum at the points nearest each other, on the two surfaces, becomes greater and greater, and increases indefinitely as their distance diminishes. It is the same with the pressure exerted by the electricity against the mass of air intercepted between the two spheres; since the pressure, as we have mentioned above, is always proportional to the square of the thickness of the electric strata. It must at last then overcome the resistance of the air, and the fluid, in escaping under the form of a spark or otherwise, must pass, previous to the contact, from the one surface to the other. The fluid thus accumulated, before the spark takes place, is of a different nature, and of nearly equal intensity on each of the spheres. If they are electrified, the one vitreously and the other resinously, it is vitreous in the first and resinous in the second; but when they are both electrified in the same manner, vitreously for example, there arises a decomposition of the combined electricity upon the sphere which contains less of the vitreous fluid than it would have in the case of contact; the resinous electricity, resulting from this decomposition, flows towards the point where the spark is preparing, and, on the contrary, the other sphere, which contains more vitreous electricity than it would have after the contact, remains vitreous over its whole extent.

The phenomena are no more the same after the two spheres have been brought in contact together, and are then removed, however little, from each other. The ratio which then exists between the total quantities of electricity with which they are charged, causes to disappear in the expression of the thickness, the term which before became infinitely great for a distance infinitely small, and no spark takes place. The electricity of the points nearest each other upon the two spheres is then very feeble, for very small distances, according to a law which calculation determines, and its intensity is nearly the same on both spheres; but when they are unequal, this electricity is vitreous on the one, resinous on the other; and it is always upon the smallest that it becomes of a nature contrary to the total electricity, which is conformable to the observations related above.

In general, all the varieties of these phenomena depend on the relation which we establish between the radii of the two spheres, and also between the quantities of electricity with which they are charged. We may even determine these proportions in such a manner, that, at a certain distance, the thickness of the electric stratum on the small sphere may be almost constant, so that this sphere may remain near the other, almost as if it were not exposed to any action, not from the weakness of the electricity on the other sphere, but in consequence of a sort of equilibrium which is then established between its action upon the smallest, and the re-action of this upon itself. In this case, the electricity diffused over the large sphere is vitreous in certain parts, resinous in others, and its thickness in different points presents very considerable variations. M. Poisson has determined the proportions of volume and of electric charge necessary to produce these phenomena; and, in this respect, as we have formerly observed, his analysis has anticipated the observations.

To complete the case of two electrified spheres placed in presence of each other, M. Poisson has calculated the changes which the greater or less distance produces on the state of the points most distant from those where the contact takes place. In this respect he has found, that, in proportion as the two spheres approach each other, the thicknesses of the electric stratum in these points tend more and more towards the values which they would have at the instant of contact. As they arrive at this limit, however, but very slowly, it hence follows, that even at very small distances, they differ yet much from what they would be if the contact or the spark actually took place. Hence we conclude also, that the spark, when it takes place at a sensible distance, changes suddenly the distribution of the electricity over the whole extent of the two surfaces, from the point where it is produced even to that which is diametrically opposite. This re-action is easily verified by experiment; we have only to fix, at certain distances from each other, along an insulated conductor, couples of linen threads, with pith balls suspended to them, and to communicate to this conductor a certain quantity of electricity, by which the threads may be made to diverge; if we then draw successively several sparks by the contact of an insulated sphere, whose volume is not too small, all the threads will be observed to be disturbed, and shaken in a manner by each explosion, in whatever part of the conductor it is produced.

For the particular case in which the two electrified spheres are removed to a great distance from each other, in relation to the radius of any one of them, M. Poisson has discovered formulae which express in a very simple manner the thickness of the electric stratum, in any point of their surfaces. We shall here state these formulae, as they enable us to explain distinctly why conducting bodies, when they are electrified, seem to attract or repel each other, although, from the manner in which electricity is distributed among them, and from its mobility in their interior, we cannot suppose that these phenomena indicate any sensible affinity which it has for their substance. Let \( r, r' \) represent the radii of the two spheres; call \( e, e' \) the thicknesses of the strata which the quantities of electricity they possess would form upon their surfaces if they were left to themselves, and exempt from all external influence, call \( a \) the distance of their centres, and place them so far from each other that the radius \( r' \) of one of them be very small compared with \( a \), and with \( a - r \). Lastly, let \( u, u' \) denote the angles formed with the distance \( a \), by the radii drawn from the centre of each sphere to any point on their surfaces; then the thicknesses \( E, E' \) of the electric stratum in these points will be expressed approximately by the following formulae.

\[ E = e + \frac{e' r'^2}{a r} - \frac{e' r'^2 (a^2 - r^2)}{r (a^2 - 2 a r \cos u + r^2)^{\frac{3}{2}}} \]

\[ E' = e' - \frac{3 e r^2}{a^2} \cos u' + \frac{5 e r^2 r'}{2 a^3} (1 - 3 \cos^2 u') \]

Here, as in the experiments of Coulomb, the angles \( u, u' \) are reckoned from the points \( A \) and \( a \) Electricity. (fig. 5), in which the surfaces of the two spheres would touch each other, if we brought them to the point of contact. The difference of symmetry in these expressions is owing to this, that the approximation from which they arise supposes the radius \( r' \), of the second, very small compared with the distance \( a - r \), which separates its centre from the surface of the other.

If it is required, for example, from these formulae to determine the state of an insulated, but not electrified sphere, which we present to the influence of another sphere charged with a certain quantity of electricity, we have only to suppose \( e' \) nothing in the equation of the second sphere, and it will then become

\[ E' = \frac{3 e\ r'^2}{a^2} \left[ \cos u' + (3 \cos^2 u' - 1) \frac{5 r'}{6 a} \right] \]

At the point \( a \), on the line \( Aa \), between the two centres, the angle \( u' \) is nothing. In this point then we have \( \cos u' = 1 \); and \( E' = - \frac{3 e\ r'^2}{a^2} \left( 1 + \frac{5 r'}{3 a} \right) \)

The thickness \( E' \), then, has always a contrary sign to that of \( e \), that is to say, that the electricity on this point, in the sphere of which the radius is \( r' \), is of a nature contrary to that which covers the sphere of which the radius is \( r \).

At the point \( d \), diametrically opposite to the preceding, the angle \( u' \) is equal to \( 180^\circ \), which gives

\[ \cos u' = -1; \text{ and } E' = + \frac{3 e\ r'^2}{a^2} \left( 1 - \frac{5 r'}{3 a} \right) \]

This value of \( E' \) has always the same sign with that of \( e \); for the factor \( \frac{5 r'}{3 a} \) is a fraction far smaller than unity, since the distance \( a \) is supposed very great, compared with the radius \( r' \); then the electric stratum will be in this point of the same nature as upon the other sphere.

Thus we see arising out of the theory the important result which we have until now only established by experiment, that while a sphere \( c \), not electrified, is placed in presence of another sphere \( C \), electrified vitreously, for example, the combined electricities of \( c \) are partly decomposed ; the resinous electricity that results flowing towards the part of \( c \) which is nearest to \( C \), and the vitreous electricity towards the part which is farthest from it.

The thicknesses of the stratum in these two points are to each other in the ratio of

\[ 1 + \frac{5 r'}{3 a} \quad \text{to} \quad 1 - \frac{5 r'}{3 a}; \]

they are nearly equal, then, since \( a \) is supposed very great in relation to \( r' \).

Hence it may be conceived that there must be upon the sphere \( c \) a series of points, in which the thickness of the electric stratum is nothing, and which form a curve of separation between the two fluids. The locus of these points will be found by putting the general expression of the thickness \( E' \) equal to zero, which gives the condition

\[ 0 = \cos u' + (3 \cos^2 u' - 1) \frac{5 r'}{6 a}. \]

If the distance \( a \) were altogether infinite, compared with the radius \( r' \), the second member of this equation would be reduced to \( \cos u' \); consequently, this cosine would be 0, which would give \( u' = 90^\circ \). The line of separation of the two fluids would then be the circumference of the great circle, of which the plane is perpendicular to the line of the centres.

But if \( a \) is not infinite, it is at least very great relatively to \( r' \). Thus, the factor \( \frac{5 r'}{6 a} \) will still be a very small fraction, and the true value of \( \cos u' \) will be equally so. We may, therefore, in calculating, neglect the product of \( \frac{5 r'}{6 a} \) by \( \cos^2 u' \), compared with the product of this same quantity by unity. With this modification the equation resolves itself and gives

\[ \cos u' = \frac{5 r'}{6 a} \]

In this case the line of separation of the two fluids is still a circle whose plane is perpendicular to the line of the centres; but the distance of this plane from the centre of the sphere, in place of being nothing, is equal to \( r' \cos u' \) or \( \frac{5 r'^2}{6 a} \), this distance being taken from \( c \) to \( a \), towards the electrified sphere \( C \).

In considering only the degree of the equation which determines generally \( \cos u' \), there would seem to be two values of this cosine which would satisfy the conditions of our problem; but it will clearly appear, that one of those roots should necessarily be greater than unity, and, consequently, will not have here any real application, as it would correspond to an arc \( u' \), which is imaginary.

When we now consider how various, how delicate, and how detached from each other, are the phenomena this theory embraces; with what exactness, also, it represents them, and follows, in a manner, all the windings of experiment, we must be convinced that it is one of the best established in physics, and that it bestows on the real existence of the two electric fluids the highest degree of probability, if not an absolute certainty. But what is not less valuable for science, it teaches us to fix, by exact definitions, the true meaning which we must attach to certain elements of the electrical phenomena, which are too often vaguely enunciated, or even confounded, with others; although the knowledge of each of them, individually, is indispensable to form a correct and general idea of the phenomena.

The first of these elements is the species, vitreous or resinous, of the electricity which exists at the surface of an electrified body, and at every point of this surface. This is determined by touching it with the proof plane, and presenting this to the needle of the electroscope, already charged with a known species of electricity.

The second element is the quantity of this electricity accumulated on every point, or, what comes to the same thing, the thickness of the electric stratum. This we still measure by touching the body with the proof plane, and communicating the electricity acquired by this contact to the fixed ball of the electric balance, the moveable one having been previously charged with electricity of the same nature. The force of torsion necessary to balance the electric reaction communicated by the plane to the fixed ball, is at equal distances proportional to the quantity of electricity which it possesses, or, what is the same thing, to the thickness of the electric stratum on the element of the surface which it has touched.

The third element which it is of importance to consider in the phenomena, is the attractive or repulsive action exerted by each element of the electric stratum upon a particle of the fluid situated at its exterior surface or beyond this surface. This attraction or repulsion is directly proportional to the thickness of the electric stratum on the superficial element which attracts or repels, and is inversely proportional to the square of the distance which separates this element from the point attracted or repelled.

In fine, the last element to be considered, and which is a consequence of the preceding ones, is the pressure which the electricity exerts against the external air in each point of the surface of the electrified body. The intensity of this pressure is proportional to the square of the thickness of the electric stratum.

By adhering strictly to these denominations, there will be no risk of falling into error from vague considerations; and if we also keep in mind the development of electricity by influence at a distance, we shall then find no difficulty in explaining all the electric phenomena.

To place this truth in its full light, we shall apply it to some general phenomena which, viewed in this manner, can be conceived with perfect clearness, but which, otherwise, do not admit but of vague and embarrassed explications. These phenomena consist in the motions which electrified bodies assume, or tend to assume, when they are placed in presence of each other, and in which they appear as if they really acted upon each other by attraction or by repulsion. But it is extremely difficult to conceive the cause of these movements, when we consider that, according to the experiments, the attraction and repulsion are only exerted between the electric principles themselves, without the material substance of the body, provided it be a conductor, having any influence on their distribution or their displacement. We cannot hence admit, that the particles of the electric principles, whatever they may be, really attract or repel the material particles of the bodies. It is absolutely necessary, therefore, that the attractive and repulsive actions of these principles, whatever they are, be transmitted indirectly to the material bodies, by some mechanism which it is of extreme importance to discover, as it is the true key to these phenomena. But we will see that this mechanism consists in the reaction produced by the resistance which the air and non-conducting bodies in general oppose to the passage of electricity.

For the sake of greater simplicity, we may first confine ourselves to the consideration of two electrified spheres A and B; the one A fixed, the other B moveable. Three cases may arise which it is necessary to discuss separately.

1. A and B non-conductors. 2. A, a non-conductor, B, a conductor. 3. A, a conductor, and B, a conductor.

In the first case, the electric particles are fixed upon the bodies A and B; by the unknown force which produces the non-conductibility. Unable to quit these bodies they divide with them the motions which their reciprocal action tends to impress upon themselves.

The forces then which may produce the motion of B, are, 1. The mutual attraction or repulsion of the fluid of A upon the fluid of B. 2. The repulsion of the fluid of B on itself. But it is demonstrated in mechanics, that the mutual attractions and repulsions exerted by the particles of a system of bodies on each other, cannot impress any motion on its centre of gravity; the effects of this internal action then destroy themselves upon each of the spheres; there cannot result from it any motion of the one towards the other; and the first kind of force, therefore, is the only one to which we need pay any attention. If the electricity is distributed uniformly over every sphere, each of them attracts or repels the other as if its whole electric mass were collected in its centre. Thus, if we call a the distance of their centres, r, r' their radii, e, e' the thicknesses of the electric strata formed upon their surfaces by the quantities of electricity introduced into them, the electric mass of each of them will be \(4\pi r^2e\), \(4\pi r'^2e'\), \(\sigma\) being the semi-circumference of which the radius is equal to unity, and the attractive or repulsive force will be expressed by

\[ \frac{16\pi^2K r^2r'^2ee'}{a^2}, \]

\(K\) being a coefficient which expresses the intensity of the force when the quantities \(a\), \(e\), \(e'\) are each equal to the unity of their species. This force transmits itself directly to the two spheres, in consequence of the adhesion by which they retain the electric particles. We see, from this expression, that the force must become nothing, if \(e\) or \(e'\) be nothing, that is, if the one of the two spheres be not primitively charged with electricity. During the motion it suffers no alteration but what arises from the distance, because the two spheres being supposed of a perfectly non-conducting substance, their reciprocal action produces upon them no new development of electricity.

In the second case, where A is a non conductor, and B a conductor, the sphere B, suffers a decomposition of its natural electricities by the influence of A. The opposite electricities which result from this decomposition unite with the new quantity which has been introduced, and dispose themselves together according to the laws of the electric equilibrium. Here the motion of B towards A may be regarded under two points of view.

Suppose, first, that without disturbing the electric equilibrium of B, we extend over its surface an insulating stratum, solid, without weight, and which may remain invariably attached to it. The electri- Electricity. city of B, unable to escape, will press as it were against this stratum, and, by this means, transmit to the particles of the body the forces by which it is urged. The forces which then act upon the system will be, 1. The mutual attraction or repulsion of the fluid of A on the fluid of B. 2. The repulsion of the fluid of B upon itself, a repulsion, however, which cannot produce any motion upon the centre of gravity of B. 3. The pressure of the fluid of B upon the insulating envelope, a pressure, again, which being exactly counterbalanced by the reaction of this coating, produces still no motion whatever. The first force, then, is still the only one to which we need pay any attention.

When the distance a, of the two spheres is very great relatively to the radii of their surfaces, the decomposed electricities of B, as we have seen at page 85, are distributed almost equally over the two hemispheres situated on the side of A, and on the opposite. In that case the actions which they suffer on the part of A are nearly equal, and destroy each other; all the force then is produced by the quantities of external electricity, \(4\pi r^2 e, 4\pi r'^2 e'\) introduced into the two spheres, which, acting as if they were wholly collected in their centres, the force becomes still \(\frac{16\pi^2 K r^2 r'^2 ee'}{a^2}\).

When the two spheres are very far from each other, the coefficient K may be considered as constant, and the attractive or repulsive force varies not but in consequence of a change in the distance a. But this is only an approximation; for, to consider the matter rigorously, the electrical state of the conducting sphere B, varies in proportion as it approaches A, on account of the separation which this produces in its natural electricities. Hence also the reciprocal action of the two spheres ought to vary in a very complicated manner, and it is probably to this that we must ascribe the error which appears in the experiments of Coulomb, at very small distances, when calculated by the simple law of the square of the distance.

The supposition of an insulating envelope without weight, serves here merely to connect the electric fluid with the material particles of the body B, and we may always regard as such the little stratum of air with which bodies are ordinarily enveloped, and which adheres to their surfaces. Yet the same result may be obtained without the aid of this intermediary; but, in that case, we must consider the pressures produced upon the air by the electricities which exist at liberty in B. These electricities, in effect, as well those that have been introduced, as those that are decomposed on it, move towards the surface of B, where the air stops them by its pressure, and prevents their escape; they dispose themselves then under this surface, as their mutual action and the influence of the body A require, resting, for this purpose, against the air, which prevents them from expanding. But, reciprocally, they press this air from within outwards, and tend to fly off with a force proportional to the square of the thickness of the electric stratum in every point. Decompose these pressures in the direction of three rectangular axes of the co-ordinates x y z, the one x being in the direction of the straight line Cc (fig. 6),


which joins the centres of the two spheres, and add together all the partial sums; it will then be found, as we shall show presently, that, in the direction of the co-ordinates y and z, they amount to nothing, and there only remains, therefore, a single resulting force, directed in the straight line Cc, that is, towards the centre of the sphere A. When the spheres are very distant from each other, compared with the radii of their surfaces, the decomposed electricities of B press the external air, in opposite directions, with a force nearly equal, and their effects destroy each other almost exactly. There only remains, then, the effect of the quantities e, e' introduced into the two spheres, and from this there results an excess of pressure in the direction of the lines of the centres,

and expressed by \(\frac{Kee'}{a^2}\), K being a constant quantity for the two spheres, that is, exactly the same as was obtained by the other method. It is evident, besides, that this expression is subject to the same limitation, since the pressures produced by the electric stratum against the external air, ought to vary with the quantity of natural electricity decomposed on B by the influence of A, in proportion as the two spheres approach each other.

The third case, in which A and B are both conductors, is resolved exactly upon the same principles, either by imagining the two electrified surfaces covered with an insulating envelope, and calculating the reciprocal actions of the two fluids which are transmitted by means of this cover to the material particles; or in considering the pressures produced on the external air by the two electric strata, and calculating the excess of these pressures in the direction of the line which joins the two centres; only, in this case, the attractive or repulsive force of these two spheres will vary in proportion as they approach each other, not only by the difference which thence arises in the intensity of the electric action, but still farther by the decomposition of the natural electricities which will be going on in the two conducting bodies A and B.

To render the mathematical exactness of these considerations evident, we shall, for the preceding case, go through the calculation of the pressures exerted against the air by the quantities of electricity introduced or developed on the two spheres. This will have, besides, the advantage of giving a new application to the formula stated at p. 84. For this purpose take, first, on the sphere A, fig. 6, any point Electricity. whatever which we may denote by M. The pressure exerted at this point against the air depends on the thickness of the electric stratum there. To know this, we must put the particular value of the angle u, which corresponds to the point M, in the general expression of E given at page 84, and multiply the square of this thickness by a constant coefficient k, which will disappear of itself, when we take the relations of the pressures among each other at different points. In this manner, the pressure for any point of either sphere calculated for the unity of the surface, will be represented by kE^2 upon the first, and kE'^2 upon the second, E, E' being taken from the formulae stated in p. 84. We shall now develope, successively, these two expressions. In the first place, as the pressure kE^2 varies from one point to another with the thickness of the electric stratum, we cannot suppose it the same, but in a very small space all round the point M, a space which must be considered as a superficial element of the sphere, and which we shall call ω; thus the expression KE^2 being calculated for the unity of surface, the pressure upon the small superficial element ω will be KωE^2. This pressure acts perpendicularly to the spherical surface A, in the direction of the radius CM; decompose it then into three others, parallel to three axes of the rectangular co-ordinates x, y, z, which have their origin at the centre C; the first, x, being in the direction of the straight line, Cc, which joins the centres of the spheres, and the two others perpendicular to this line. To effect this decomposition, we must multiply the normal pressure KE^2 by the cosines of the angles which the radius CM forms with the co-ordinates x, y, z; that is, by

\[ \frac{x}{r}, \frac{y}{r}, \frac{z}{r}, \]

since, in the formula of p. 84, we have

represented by r the value of the radius CM of the sphere A. We will thus have the three following component parts

parallel to the co-ordinates x; \( K \frac{x}{r} \omega E^2 \)

parallel to the co-ordinates y; \( K \frac{y}{r} \omega E^2 \)

parallel to the co-ordinates z; \( K \frac{z}{r} \omega E^2 \)

But we must observe, first, that it is absolutely of no use paying any attention to the two last, because the efforts which each of them makes, on the whole extent of the surface, mutually destroy each other, on account of the symmetrical disposition of the electricity relatively to the axis of the co-ordinates x, which joins the two centres. If we consider, in effect, the force,

for example, \( K \frac{z}{r} \omega E^2 \) for the point M, situated in the figure under the plane of the co-ordinates xy; we shall find, above this plane, another point M' situated quite similarly, and of which the co-ordinates x, y, z, will consequently be the same, with this only difference, that z will there be negative, on account of its opposite situation relative to the origin of the co-ordinates.

For this second point, the element ω, and the pressure KE^2, will be also absolutely the same; ω on account of the symmetry of the surface of the sphere A; E^2 on account of the symmetrical disposition of the electricity round the axis of the co-ordinates x, which joins the centres of the two spheres A and B; but the component force which proceeds parallel to the co-ordinates z, will be — \( K \frac{z}{r} \omega E^2 \), on account of the negative sign of z; this force and its analogous one,

+ \( K \frac{z}{r} \omega E^2 \) being equal, and in opposite directions,

will mutually destroy each other, and a similar equilibrium will be equally obtained, in this kind of pressure, for all the other couples of points M, M', which correspond on the two sides of the plane of xy.

A similar process of reasoning will prove that the forces \( K \frac{y}{r} \omega E^2 \) will destroy each other two and two, upon corresponding points, taken on the two sides of the plane of xz, and of which the co-ordinates will be + x, + y, + z for the one, and + x, — y, + z for the other.

It remains, then, to consider the components of the pressures, taken parallel to the co-ordinates x; that is, parallel to the straight line which joins the centres Cc, of the two spheres; and, indeed, from the symmetrical disposition of electricity round this straight line, it is evident that it cannot have any motion but in this single direction; and, consequently, these components alone must produce the tendency of the two spheres towards each other.

To obtain, in the simplest manner, the sum of all these components, it must be remarked, that their general expression \( K \frac{x}{r} \omega E^2 \) contains no variable but x; for cos. u, which enters into the value of E^2 is equal to \( \frac{x}{r} \); it hence follows, that their intensities are equal in the points relatively to which the co-ordinate x is the same, and which are consequently situated upon one small circle, parallel to the plane of the co-ordinates y, z. Besides, as all these points are equally distant from the line of the centres, it is clear that the total result of the equal forces which are applied to them will be in the direction of this line; consequently this will also be the direction of the general result of all the efforts of this kind exerted upon the whole surface of the sphere A.

To obtain now, easily, the sum of all these forces, parallel to the line of the centres, which is here that of the co-ordinates x, let us begin by joining together the values of x which are equal and contrary; for the thicknesses E, of the electric stratum on the two hemispheres of A being almost equal, from the supposition that the two spheres are very distant, the pressures corresponding to opposite values + x and — x, must be almost equal also; and, as the components which they give parallel to the co-ordinates x are in a contrary direction, their sum must be reduced to a very small quantity. To introduce this circumstance, call E, what E becomes when we change \(+x\) into \(-x\); then the expressions of the corresponding components parallel to the co-ordinates \(x\) will be,

On the side of the positive co-ordinates \(x\), \[ + K \frac{x}{r} \omega E^2 \] Tending to move the air in the direction AB

On the side of the negative co-ordinates \(-x\), \[ - K \frac{x}{r} \omega E^2 \] Tending to move the air in the direction BA

We preserve the superficial element \(\omega\) always of the same value, because it is exactly alike in the two cases, on account of the symmetrical form of the sphere on the two sides of the plane of the co-ordinates \(yz\); adding these two components to each other with their actual sign, their sum will express the element of the total resulting force which tends to carry the air in the direction AB. This, then, will be \(K \frac{x}{r} \omega (E^2 - E_1^2)\), or, what is the same thing, \[ K \frac{x}{r} \omega (E + E_1)(E - E_1) \]

But, from the formula of p. 84, we have generally \[ E = e + \frac{e' r'^2}{ar} - \frac{e' r'^2 (a^2 - r^2)}{r(a^2 - 2ar \cos u + r^2)^{\frac{3}{2}}} \] To change \(+x\) into \(-x\), \(+ \cos u\) must become \(- \cos u\), because \(\cos u = \frac{x}{r}\); for this second case, then, we shall have \[ E_1 = e + \frac{e' r'^2}{ar} - \frac{e' r'^2 (a^2 - r^2)}{r(a^2 + 2ar \cos u + r^2)^{\frac{3}{2}}} \] consequently, by subtracting these equations from each other, we have \[ E - E_1 = \frac{e' r'^2 (a^2 - r^2)}{r} \left\{ \frac{1}{(a^2 + 2ar \cos u + r^2)^{\frac{3}{2}}} - \frac{1}{(a^2 - 2ar \cos u + r^2)^{\frac{3}{2}}} \right\} \] or, what is the same thing, and is better adapted for approximations, \[ E - E_1 = \frac{e' r'^2}{ar} \left( 1 - \frac{r^2}{a^2} \right) \left\{ \frac{1}{\left( 1 + \frac{2r}{a} \cos u + \frac{r^2}{a^2} \right)^{\frac{3}{2}}} - \frac{1}{\left( 1 - \frac{2r}{a} \cos u + \frac{r^2}{a^2} \right)^{\frac{3}{2}}} \right\} \] Since we suppose the two spheres very distant from each other, compared with the magnitude of their radii, \(\frac{r}{a}\) will be a very small fraction; hence we may develope this expression for \(E - E_1\) into a converging series of the ascending powers of \(\frac{r}{a}\), this will be effected by the binomial theorem; and taking only the first power of \(\frac{r}{a}\), which, in the case we are considering, will contain an infinitely great proportion of the total result, compared with the other powers, it will become \[ E - E_1 = \frac{e' r'^2}{ar} \left\{ 1 - \frac{3r}{a} \cos u - 1 - \frac{3r}{a} \cos u \right\} \] or by reduction \(E - E_1 = -6r'^2 e' \cos u\).

This value of \(E - E_1\) must now be multiplied by \(E + E_1\) to form the factor \(E^2 - E_1^2\) which enters into the expression of the total resulting force; but since \(E - E_1\) is already of the order \(\frac{r'^2}{a^2}\) it is evident that, in \(E + E_1\), we must confine ourselves to the terms which are not divided by \(a\); this limitation reduces the value of \(E + E_1\) to \(2e\), and employing this to multiply \(E - E_1\) there results \[ E^2 - E_1^2 = \frac{-12ee'r'^2}{a^2} \cos u. \] It only remains to substitute this value in the expression of the resulting force, parallel to the co-ordinates \(x\), which we have found equal to \(K \frac{x \omega}{r} (E^2 - E_1^2)\) for the superficial element \(\omega\); and by putting for \(\frac{x}{r}\) its value \(\cos u\), the expression will become \[ \frac{-12Ke'e'r'^2}{a^2} \omega \cos^2 u. \] Each of these partial results is proportional to the superficial element \(\omega\), and to the square of the cosine of the angle, which these elements form with the axis of the co-ordinates \(x\). But, if we compare them together upon different spheres, this angle will always be expressed by the same values; for the equal values of \(u\), however, the superficial element \(\omega\) will vary in magnitude proportionally to the square of the radius \(r\) of the sphere. Consequently, the sum of all the values of the factor \(\omega \cos^2 u\), extended to every sphere, will only vary from each other in the ratio of the square \(r^2\); it may be represented then by \(K'r^2\), \(K'\) being a constant and numerical coefficient which may be found, and which, in reality, is found by the processes of the integral calculus. Supposing it known then, the total result of the pressures parallel to the co-ordinates \(x\) will be \[ \frac{-12KK'e'e'r'^2}{a^2} \] (1). It will be directly proportional then to the quantities \(4\pi r'^2 e\), \(4\pi r'^2 e'\) of external electricity which they possess, and inversely proportional to the square of the distance of the two centres. When the quantities of electricity given to the two spheres are of the same nature, whether vitreous or resinous, the values of \(e\) and of \(e'\) must be considered as having the same sign. In that case the expression (1) is negative, that is to say, according to what has been previously admitted, that, in this case, the air which surrounds the sphere Electricity. A, is more pressed in the direction BA, than in the direction AB. It will not then press equally the sphere A, as it did before it was electrified; it will press it less on the side which is most distant from the other sphere, since it is in that direction that the electric reaction is the strongest. Consequently, if the sphere A is at liberty to move, and deprived of its weight, or if its weight be sustained by a thread of suspension, it will put itself in motion from the side where the atmospheric pressure has become the weakest, that is to say, that it will recede from the other sphere B.

The contrary would happen, according to our formula, if the quantities e,e' of electricity introduced into the two spheres were of a different nature, for then it would be necessary in the calculation to give to them different signs. The formula (1), which represents the total result of the pressures exerted against the air parallel to the line of the centres, will then become positive, that is to say, according to what has been already agreed on, the external air will be more pressed in the direction AB, than in that of BA; the sphere A will move then in the direction in which the external pressure will have become the weakest, that is towards the sphere B, agreeably to observation.

We have hitherto only considered the effect of the pressures round the sphere A, but the same reasonings and calculations will apply to the other sphere; only we must then employ, instead of E and E', the expressions of the electric strata which correspond to them, and which we have seen to be

\[ E' = e' - \frac{3er^2}{a^2} \cos u' + \frac{5er^2r'}{2a^3} (1 - 3 \cos^2 u') \]

It might be shown for this case as well as for the other, that there cannot be any inequality of pressure but in the direction of the co-ordinates x; then comparing the points of the surface which correspond to the two sides of the plane of the co-ordinates y z, we shall find that the element of the resulting force of the pressures parallel to the co-ordinates x, is in general expressed by \( \frac{Kxw'}{r'} (E'^2 - E_1'^2) \) in representing by

\( E' \) what \( E' \) becomes when we change \( + x \) into \( - x \), that is to say, \( + \cos u' \) into \( - \cos u' \), the angle \( u' \) being here reckoned from the point a, situated on the side of A upon the line of the centres. By next considering the spheres as very distant we will obtain in the same manner the value of \( E' - E_1' \). Approximating no farther than the first power of \( \frac{r'}{a} \), this will give \( E' - E_1' = \frac{-6er^2}{a^2} \cos u' \). We have then only to take \( E + E_1' = 2e' \), and putting these values in the expressions of the partial resulting force, it will become \( \frac{-12Keer^2w' \cos^2 u'}{a^2} \). It may be demonstrated as above, that the sum of the factors \( w' \cos^2 u' \) will be proportional to the square of the radius of the sphere B, and may besides be represented by \( K'r'^2 \); \( K' \) being the same numerical coefficient we have already employed. For the total Electricity, resulting force then, the expression will finally become

\[ -12KK'e^r r'^2 / a^2, \]

that is, exactly the same which we have obtained for the other sphere, which ought to be the case, since in these sort of phenomena action and reaction are always equal. Here, as in the example immediately above, the positive sign of the expression will signify that the resulting force of the pressures exerted against the air round the sphere B, is directed towards the other sphere, and the negative sign will signify that this resulting force is directed the opposite way. The first case will take place when the electric charges ee' are of a contrary nature; in that case, the sphere B will advance from the side where the atmospheric pressure is weakest, that is towards A; the other case will happen when the electric charges ee' are of the same nature, then B will recede from A.

The common expression for the result of the pressures vanishes for both the spheres, when e or e' is nothing, that is, when one of them is in the natural state. This seems to indicate that they would then neither approach nor recede from each other, while, in reality, we know that in this case they always approach. This apparent contradiction is owing to the degree of approximation at which we stopped our developement of the above expression. We have supposed our two spheres very distant from each other, compared with the radii of their surfaces; the result of this is, that whatever be the quantity \( 4\pi r^2e, 4\pi r'^2e' \) of external electricity which we have introduced into each of them, it will distribute itself almost uniformly over the two hemispheres, anterior and posterior; so that the difference of the pressures exerted against the air by these two hemispheres, which is the only cause of motion, will be very small, and it is to this degree of minuteness that we have confined our approximations in developing \( E'^2 - E_1'^2 \).

If, however, the one of the two spheres, B for example, is only electrified by the influence of the other, which we always suppose very distant, the developement of its natural electricities will be still very feeble, and of the same order of minuteness with that to which we have confined our approximations; but this weak electricity still dividing itself between the two hemispheres of B, in a manner nearly equal, as in the example immediately above, the difference of pressures round the two hemispheres will become very minute in a still lower degree—will become a quantity of the second order of minuteness, and consequently, cannot be found in our developements, such as we have limited them. To obtain it complete, we must not, in the calculation of \( E' + E_1' \), confine ourselves to quantities, independent of \( \frac{r'}{a} \), but take its whole value. We will then have, first of all,

\[ E' = e' - \frac{3e r^2}{a^2} \cos u' + \frac{5e r^2 r'}{2 a^3} (1 - 3 \cos^2 u') \]

then changing \( + x \) into \( - x \), or \( + \cos u' \) into \( - \cos u' \), we will have, E' = e' + \frac{3\,e\,r^2\,\cos.\,u'}{a^3} + \frac{5\,e\,r^2\,r'}{2a^3}(1-3\cos.^2u')

then adding these two expressions,

\[ E' + E'_1 = 2\,e' + \frac{5\,e\,r^2\,r'}{a^2}(1-3\cos.^2u') \]

This complete value of \( E' + E'_1 \) will now no more vanish when \( e' \) is nothing, but it will be seen that the terms which remain are of the order of those which we have neglected in our first approximation. Making, then, here \( e' \) equal to nothing, it remains,

\[ E' + E'_1 = \frac{5\,e\,r^2\,r'}{a^3}(1-3\cos.^2u') \]

we also find, as before,

\[ E' - E'_1 = -\frac{6\,e\,r^2}{a^2}\cos.\,u' \]

with these values, the expression of the partial resulting force

\[ \frac{K\,\omega'\,r'}{r^3}(E''_{r'} - E''_{r}), \text{ or } K\,\omega'(E''_{r'} - E''_{r})\cos.\,u' \]

becomes

\[ -\frac{30\,K\,e^2\,r^4\,r'}{a^5}\cdot\omega'\cos.^2u'\,(1-3\cos.^2u'). \]

It only remains to take the sum of it over all the extent of the surface of the sphere B; but, in this operation, the variable factor, \( \omega'\cos.^2u'\,(1-3\cos.^2u') \), will give a result proportional to the square of the radius \( r' \) of the sphere B, and which we may consequently represent by \( K''r'^2 \), \( K'' \) being a constant numerical co-efficient different from \( K' \); thus, the total resulting force will at last be

\[ -\frac{30\,KK''\,e^2\,r^4\,r'^3}{a^5}. \]

This force, then, will be of an order of minuteness, much inferior to that which we obtained at first, when \( e' \) was not supposed to be nothing, since the radii there are divided by the fifth power of the distance of the centres of the two spheres, instead of the simple square which we had in the other approximation. It is obvious, that experiments of this kind, made with the electric balance, by charging only one of the balls, might produce an error as to the true law of the phenomena, if the theory did not throw light upon them; for one might be led to conclude from them, that the apparent attraction determined in this case is not reciprocally as the square of the distance of the centres of the two spheres, which nevertheless would be contrary to the truth; consequently, when it is meant to put this simple law of the square to the test, the balls must not be allowed to approach so near to each other, that the electricity developed by their reciprocal influence may bear any sensible proportion to the quantities of external electricity introduced into them; and this is the reason, that, in these experiments, it is always more certain to employ, instead of balls, small circular disks of gilt paper; for, on account of the thinness of these disks, the quantities of vitreous or resinous electricity developed at their surfaces, having nearly no room to separate from each other, their actions on the exterior bodies must be always about exactly alike, and cannot, therefore, alter the results any more than if their development had not taken place.

The theory, which we have thus explained in regard to spheres, applies equally well to bodies of any form whatever; but here the difficulty of the analysis prevents us from anticipating any thing but the general effects which the different pressures produce, without our being able to reduce them to numbers. Those who wish to enter into the details, will find them in Biot's Physique Mathematique, Tom. II. Here it will suffice to have established the general mode of reasoning applicable to all the questions of this kind, and to have followed out the whole development for the single case, which analysis has been able as yet completely to surmount. We shall add, that before the theory had acquired its actual precision, it could not be clearly conceived how the attractions and repulsions, which in reality only take place between the electrical principles themselves, were communicated to the material particles of the electrified bodies; and philosophers were reduced to the necessity of denoting this effect by the vague word tension, which represented the electricity like a kind of spring placed between the bodies, and tending to make them approach or recede.

In the preceding observations, we have only attended to the statics and the dynamics of electricity, that is, to the laws of its equilibrium and of its motion. To complete these additions, it would be necessary still to consider its chemical action, concerning which many discoveries have also been made since the publication of the Encyclopædia; but it will be better to place these results under the article Galvanism, as they will there be united with a great number of others, which will mutually throw light upon each other, and at the same time will render more certain, and more general, the theoretical consequences which may be drawn from them. We shall only present here some necessary modifications of the ideas which may be formed from the Encyclopædia, in regard to the nature of the two electricities, and to the physical impressions by which their material existence seems to become manifest to our senses. The light which the electric spark excites in the air, does not by any means prove, that the electric principle itself is luminous, any more than the phosphoric odour, which the electrified points produce on the organ of smell, proves that this principle is odorous. It is now known by experiment, that every sudden stroke, every rapid motion, impressed on a mass of air, which cannot yield with sufficient agility, excites in it a degree of light; and, in order that it may thus be excited in the open air, it is sufficient that it be impelled more vigorously than its own resistance permits it to give way. Whatever may be the nature of electricity, we know that it produces upon bodies in which it is contained, and on those which it traverses, a repulsive force, which becomes, in many instances, extremely energetic. We know, besides, that, without any estimable mass for our most sensible balances, it may yet impress upon these bodies very rapid motions, when we accumulate it, and dispose it in such a manner, that the pressures which it exerts against the air around these bodies cannot mutually balance each other. Hence Electricity, we may conclude, by the laws of mechanics, that the velocity of its transmission must be immense; but with an excessive velocity, and a very great repulsive force, what more is wanting to compress the air, and even the rarest vapours, even to the point where the disengagement of light begins? This idea, which we believe was first proposed by M. Biot (Annales de Chimie, 1805, Tom. LIII. p. 321), seems to us to give a very plausible reason for these phenomena; and the possibility of so simple an explanation, is enough for authorizing us to reject as hypothetical the conclusion which has been too often drawn from this phenomenon, that the electric principle is heat, or a modification of heat. There is still less occasion to insist on the uncertainty of that other hypothetical conclusion, that the electric principle is odorous. Whenever it is discovered that this principle may occasion mechanical motions in lifeless bodies, and that its rapid transmission may excite muscular contractions in organised bodies, by a moment's reflection we can conceive that the same influence may excite tickling in the pituitary membrane, in such a manner as to produce in us a sensation like what the impression of a body really odorous may occasion. These phenomena, then, are not characters peculiar to the electric principles, and essential to their existence; they are very probably simple modifications which its repulsive force produces in bodies, or in us. We may venture to add, that they do not prove even its materiality; for what proof is there, that a repulsive force cannot be excited, or arise in any point of space, without being attached to sensible and ponderable particles? The only strong induction, the only one perhaps of really any weight as to the materiality of the electric principles, is, that in all the phenomena of their equilibrium and motions, they act exactly as two fluids would do, whose particles would mutually repel each other, and attract those of the other fluid reciprocally as the squares of the distances. This constitution being supposed, all the electrical phenomena become the rigorous mechanical consequences of it; can be anticipated with the most perfect precision, and can even be reduced to numbers in their minutest details, as well as in their most intricate windings, when the analysis may apply to them. Does not this, then, afford a very strong presumption, that these two principles are in effect and in reality such as this constitution, deduced from the phenomena, indicates?

Finally, Whatever the true nature of electricity may be, in itself, since the constitution which we have ascribed to the two electric fluids reproduces exactly and numerically all the phenomena, which as yet it has been possible to develope by calculus, this is enough to entitle us to admit this constitution in our farther inquiries. For we are authorized to conclude, from the verifications already established, that the real nature of electricity, whatever it may be, must conform itself to the facts with equal exactness; and, on the other hand, that, when applied to these facts, it will reduce itself to the same conditions which we have attributed to the two fluids; so that the facts will not flow from it otherwise, nor by other formulas, than what we now employ. But new observations, and even new applications, will serve, when the mathematical analysis will have become more perfect, to confirm or destroy the physical reality of the theory, and will show whether it is an exact and general interpretation of all the phenomena, or whether only an approximate and particular expression of those which till now it has embraced.

This progression of ideas may be already observed in the succession of the theories anterior to this, which we have here explained. And among these speculations one has been too justly celebrated, and too useful, to be passed over in silence. Most of the electric phenomena, if we look only to their general circumstances, may be represented, by supposing only the existence of a single electric fluid, of which a certain quantity is diffused through all bodies, and forms their natural state. The excess of this fluid in bodies produces what we have called the vitrous electricity; its deficiency what we have called the resinous electricity. Hence arise two states of the bodies, which the followers of this system denote by the terms positive and negative. They admit, also, that the particles of the electric fluid mutually repel each other; but as experience shows, that bodies in the natural state do not exert any action on each other, they are besides constrained to suppose, that the electric particles are attracted by the matter itself of the bodies; and lastly, as it is proved by a profound discussion and calculation, that even this condition will not be sufficient to establish the equilibrium, it becomes necessary farther to admit, that the particles of all bodies exert on each other a repulsive action, sensible, like the electric influences themselves, at great distances, and varying according to the same laws with the distance. Franklin, who first imagined this system, and who employed it ingeniously to unite all the electric phenomena known in his time, and which till then were scattered and unconnected together,—Franklin did not perceive the consequence to which his hypothesis led. Aepinus was the first who, by an exact analysis of all the forces by which the electric equilibrium was brought about, discovered the necessity of a repulsion between the particles of the bodies (Tentamen Theoriae Electricitatis et Magnetismi, p. 39), and, after him, the celebrated philosopher Henry Cavendish was led to the same consequence; for he made this repulsion one of the fundamental conditions of his hypothesis on the nature of electricity, published in the Philosophical Transactions for the year 1771, and which accords exactly with the hypothesis of Aepinus.

Although the existence of such a repulsive force between material particles of all bodies may, at first view, seem quite opposite to the general phenomena of the universe, particularly to the great law of celestial attraction, it is not so in fact; for this repulsion, as it is employed by Aepinus and Cavendish, would be exactly balanced by the mutual attraction which the hypothesis supposes to exist between the electric fluid and all material substances, when they are in their natural state of electricity; so that no resulting force would be exerted in this state by their mutual actions; and then, all other properties, or forces emanating from the bodies, as attraction seems to be one, could equally exist and manifest their power in the state of electrical saturation, as well as if there existed no repulsive action whatever between the matter of bodies. In effect, if we admit such a state of things, we may connect together by it a great number of electric phenomena, conceive their mutual dependence, and anticipate them, not, indeed, as to the quantity and number, but in their general circumstances. This view of the subject is fully discussed in the Encyclopaedia, where the results of this system are explained in detail. We thus explain, for example, the attractions and repulsions of electrified bodies, and even the development of electric properties in bodies, in the natural state, by the sole influence at a distance of an electrified body. But, from the time when this theory was first imagined, many circumstances of the phenomena have been more accurately, and more precisely fixed, and many have been limited by exact measurements. In fine, we know them by numbers; and it is in number that the theory must now represent them. When Æpinus and Cavendish wrote, the law of electric attractions and repulsions was not yet ascertained by experiment. It could be doubted, then, whether these forces varied according to the cube, to the square, or to some other power of the distances; and it was consequently impossible to compute numerically the distribution of electricity in the bodies where it disposes itself by its proper equilibrium, or to assign the proportion of its allotment between two bodies of a given form, since these delicate effects are dependent on the laws by which the action of the fluid upon itself and other bodies is regulated. In the law of the cube, for example, the distribution and the division of electricity would be different from what they are in the law of the square, and the former may be now rejected, as giving consequences contrary to actual observation. In like manner, if we now introduce the law of the square in Æpinus's or Cavendish's hypothesis, we would probably be led to consequences which would be found inconsistent with the exact measurement of these phenomena which we now possess; but this deduction has not yet been made, and seems very difficult to be done. Happily it does not now appear to be of any importance, since the hypothesis so pursued could have no greater success than to agree with the facts, which the theory of the two fluids already does in the most exact manner, and what is of no small consideration, with a complete evidence, simplicity, and facility to be expressed by analysis.

(2. z.)

ELLiptic Turning. Wood and other substances are turned into an elliptic form by means of a chuck, which is applied on the common turning lathe. This chuck is on the principle of the trammel, Plate LXXIX. fig. 1. The grooves in the chuck are much wider than in the trammel, and the points of the chuck that correspond to C, D, of the trammel, fig. 1, remain fixed, and in one horizontal line, whereas in the trammel it is these points that are put in motion.

If two straight lines, crossing each other, be drawn on a piece of transparent paper or mica; and if this transparent paper be laid upon a sheet of white paper, with two points marked on it; and if the transparent paper be moved round, so that the cross lines shall travel over the two points, in like manner as the two points, C, D, in the trammel, fig. 1, travel over the cross grooves of the trammel; and if the point of a pencil be held fixed, and touching the transparent paper, so as to leave a trace on the transparent paper, when the paper is moved; then after the transparent paper has made a revolution, the trace left on it by the point of the pencil is an ellipse. This method of describing an ellipse represents the action that takes place in the chuck for turning ellipses; the point of the pencil which remains unmoved is in the same situation as the turner's gouge; the transparent paper, which receives the trace of the ellipse from the fixed pencil, is analogous to the wood, which is to be turned into the form of an ellipse by the fixed cutting gouge.

In fig. 2, the chuck is represented as fitted on a common turning lathe, of which A is the pulley of the maundrel, B and C are the sides of the frame supporting the pulley, P the rest, D the frame in which the rest slides, E F the feet of that frame, I the nut and screw which serve to fix the rest, G H are the continuation of the sides B C. K is the elliptic chuck, with two grooves, through which the knobs of the slider pass; these knobs are connected by a strong bar of iron screwed into their ends, and on this bar of iron is seen the screw for fastening the board, to which is fixed the wood or other substance which is to be turned elliptically.

Fig. 3. shows the other side of the chuck, which, in fig. 2, is turned towards the side of the frame C. N, in fig. 3, is the board with the grooves, which contain the slider O. In the middle of N is seen the end of the screw, which is fixed to the maundrel. The board N has a circular motion, being fixed on the axis of the maundrel, whilst the slider O, at the same time that it is carried round by the circular motion of N, is constrained to perform other motions by the grooves in N, and by the groove in O, fig. 3, which slides on the ring M, fig. 4.

In fig. 4, L is a part of the side C of the maundrel frame, with the ring M fastened to it. On this ring, the broad groove in the slider O, fig. 3, moves when the lathe is set a going; and this groove is at right-angles to the grooves in N, fig. 3, in which the knobs of O move. In fig. 4, it is seen that the centre of the ring M may be made to coincide with the centre of the spindle of the maundrel, in which case a circle is described. If the ring is fixed, so that the centre of the ring does not coincide with the centre of the maundrel, an ellipse is described by the wood screwed upon the bar of O, in fig. 2; and the most eccentric ellipse that the machine describes is obtained, when the maundrel is at the circumference of the ring. The centre of the spindle of the maundrel, and the centre of the ring M, are always in one immovable horizontal line, and are analogous to the points C, D, of the trammel, fig. 1. In fig. 3, it is seen that the sides of the grooves may be brought nearer together by means of screws, so that the sliders and the cylindric ring may fit exactly to the grooves. The best elliptic chucks are made of brass. See Mechanical Exercises, by Peter Nicholson, London, 1812.