Since by the hypothesis of incompressibility, p is constant,

pV + p =C,

where C is a constant; and if we distinguish by suffixes the symbols belonging to the two ends of the canal where it meets the bodies 4, and A

pV1 +P1 = p V2 + P3•

1

2

But we have seen that p1 = p2 = 0. Hence dividing by p we find for the condition of equilibrium

or the electric potential of the two bodies must be equal.

We arrive at precisely the same condition if we suppose the bodies connected by a fine wire which is made of a conducting substance.

Let Vas before be the potential at any given point due to the electrified bodies, and let V, be its value in A,, and its value in A ̧, and let' be the potential due to the electrification of the wire at the given point, then the condition of equilibrium of the electricity in the wire is that VV' must be constant for all points within the substance of the wire. Hence at the two ends of the wire

Hence the actual potential due to the bodies and the wire together is the same in A, and A ̧.

The only difference, then, between the actual case of the wire and the hypothetical case of the canal is that the surface of the wire is charged with electricity in such a way as to make its potential everywhere constant, whereas the canal is exactly saturated, and the effect of variation of potential is counteracted by variation of pressure.

Hence the canal produces no effect in altering the electrical state of the other bodies, whereas the wire acts like any other body charged with electricity.

The charge of the wire, however, may be diminished without limit by diminishing its diameter. It is approximately inversely proportional to the logarithm of the ratio of a certain length to the diameter of the wire. Hence by making the wire fine enough, the disturbance of the distribution of electricity on the bodies may be made as small as we please.

From the Preface to Green's "Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism."

"CAVENDISH, who having confined himself to such simple methods. as may readily be understood by any one possessed of an elementary knowledge of geometry and fluxions, has rendered his paper accessible

to a great number of readers; and although, from subsequent remarks, he appears dissatisfied with an hypothesis which enabled him to draw some important conclusions, it will readily be perceived, on an attentive perusal of his paper, that a trifling alteration will suffice to render the whole perfectly legitimate.

In order to make this quite clear, let us select one of Cavendish's propositions, the twentieth for instance [Art. 71], and examine with some attention the method there employed. The object of this proposition is to show, that when two similar conducting bodies communicate by means of a long slender canal, and are charged with electricity, the respective quantities of redundant fluid contained in them will be proportional to the n-1 power of their corresponding diameters; supposing the electric repulsion to vary inversely as the n power of the distance.

This is proved by considering the canal as cylindrical, and filled with incompressible fluid of uniform density: then the quantities of electricity in the interior of the two bodies are determined by a very simple geometrical construction, so that the total action exerted on the whole canal by one of them shall exactly balance that arising from the other; and from some remarks in the 27th proposition [Arts. 94, 95] it appears the results thus obtained agree very well with experiments in which real canals are employed, whether they are straight or crooked, provided, as has since been shown by Coulomb, n is equal to two. The author, however, confesses he is by no means able to demonstrate this, although, as we shall see immediately, it may very easily be deduced from the propositions contained in this paper.

For this purpose let us conceive an incompressible fluid of uniform density, whose particles do not act on each other, but which are subject to the same actions from all the electricity in their vicinity, as real electric fluid of like density would be; then supposing an infinitely thin canal of this hypothetical fluid, whose perpendicular sections are all equal and similar, to pass from a point a on the surface of one of the bodies through a portion of its mass, along the interior of the real canal, and through a part of the other body, so as to reach a point A on its surface, and then proceed from A to a in a right line, forming thus a closed circuit, it is evident from the principles of hydrostatics, and may be proved from our author's 23rd proposition [Art. 84], that the whole of the hypothetical canal will be in equilibrium, and as every particle of the portion contained within the system is necessarily so, the rectilinear portion ad must therefore be in equilibrium.

This simple consideration serves to complete Cavendish's demonstration, whatever may be the form or thickness of the real canal, provided the quantity of electricity in it is very small compared with that contained in the bodies.

An analogous application of it will render the demonstration of the 22nd proposition [Art. 74] complete, when the two coatings of the glass plate communicate with their respective conducting bodies by fine metallic wires of any form."

NOTE 4, ART. 83.

On the charges of two equal parallel disks, the distance between them being small compared with the radius.

The theory of two parallel disks, charged in any way, may be deduced from the consideration of two principal cases.

The first case is when the potentials of the two disks are equal. If the distance between the disks is very small compared with their diameter, we may consider the whole system as a single disk, the charge of which is approximately the same as if it were infinitely thin. Hence if V be the potential, and if we write A for the capacity of the first disk, and B for the coefficient of induction between the two disks, the charge of the first disk is

The second case is when the charges of the disks are equal and opposite. The surface-density in this case is approximately uniform except near the edges of the disks. I have not attempted to ascertain the amount of accumulation near the edge except when n = = 2. If we suppose the density uniform, then for a charge of the first disk equal to a2, its potential, when b the distance between the disks is small compared with a the radius, will be approximately

In this case, however, we can carry the approximation further, for it is shown in Note 20 that

It is shown in "Electricity and Magnetism," Art. 202, that when two disks are charged to equal and opposite potentials, the density near the edge of each disk is greater than at a distance from it, and the whole charge is the same as if a strip of breadth had been added all round the disk.

b

2π

This proposition seems intended to justify those experimental methods in which the potential of the earth is assumed as the zero of potential.

Cavendish, by introducing the idea of degrees of electrification, as distinguished from the magnitudes of overcharge and undercharge, very nearly attained to the position of those who are in possession of the idea of potential. But the very form of the phrases "positively or negatively electrified," which Cavendish uses, confers an importance on the limiting condition of "no electrification," which we hardly think of attributing to "zero potential." For we know that all electrical phenomena depend on differences of potential, and that the particular potential which we assume for our zero may be chosen arbitrarily, because it does not involve any physical consequences.

It is true that the mathematicians define the zero of potential as the potential at an infinite distance from the finite system which includes

the electric charges. This, however, is not a definition of which the experimentalist can avail himself, so he takes the potential of the earth as a zero accessible to all terrestrial electricians, and each electrician "makes his own earth."

The earth-connexion used by Cavendish is described in Art. 258. But when the whole apparatus of an electrical experiment is contained in a moderate space, such as a room, it is convenient to make an artificial "earth" by connecting by metal wires the case of the electrometer with all those parts of the apparatus which are intended to be at the same potential, and calling this potential zero.

It appears by observation, that in fine weather the electric potential at a point in the air increases with the distance from the earth's surface up to the greatest heights reached by observers, and in all parts of the earth. It is only when there are considerable disturbances in the atmosphere that the potential ever diminishes as the height increases. Hence the potential of the earth is probably always less than that of the highest strata of the atmosphere.

If the earth and its atmosphere together contain just as much electricity as will saturate them, and if there is no free electricity in the regions beyond, then the potential of the outer stratum of the atmosphere will be the same as that at an infinite distance, that is, it will be the zero of the mathematical theory, and the potential of the earth will be negative.

NOTE 6, ART. 97, p. 43.

On the Molecular Constitution of Air.

The theory of Sir Isaac Newton here referred to is given in the Principia, Lib. II., Prop. XXIII.

Newton supposes a constant quantity of air enclosed in a cubical vessel which is made to vary so as to become a cube of greater or smaller dimensions. Then since by Boyle's law the product of the pressure of the air on unit of surface into the volume of the cube is constant; and since the volume of the cube is the product of the area of a face into the edge perpendicular to it, it follows that the product of the total pressure on a face of the cube into the edge of the cube is constant, or the total pressure on a face is inversely as the edge of the cube.

Now if an imaginary plane be drawn through the cube parallel to one of its faces, the mutual pressure between the portions of air on opposite sides of this plane is equal to the pressure on a face of the cube. But the number of particles is the same, and their configuration is geometrically similar whether the cube is large or small. Hence the distance between any two given molecules must vary as the edge of the