when n = 2, λ = A, or the density is uniform.

Since the fluid is in equilibrium in all these cases, the potential is uniform throughout the body. We may therefore determine the value of the potential at any point within the body by finding its value at any selected point, as for instance at the centre. If de be an element of the fluid, and its distance from the given point, the corresponding

element of the potential due to the force whose value is er

We thus find for the potential of the sphere

1

is

n- 1

When n becomes equal to 4, V becomes infinite.

When n is equal to 2, V = Qa ̄'.

For the plate bounded by parallel planes, V is infinite, except for values of n between 3 and 4, for which

where σ。

is the quantity of fluid in unit of area of the plate.

NOTE 3, ART. 69.

On canals of incompressible fluid.

It appears from several passages (Arts. 40, 236, 273, 276, 278, 294, 348) that Cavendish considered that the weakest point in his theory was the assumption that the condition of electric equilibrium between two conductors connected by a fine wire is the same as if, instead of the wire, there were a canal of incompressible fluid defined as in Art. 69.

It is true that the properties of the electric fluid, as defined by Cavendish in Art. 3, are very different from those of an incompressible fluid. But it is easy to show that the results deduced by Cavendish from the hypothesis of a canal of incompressible fluid are applicable to the actual case in which the bodies are connected by a fine wire.

In what follows, when we speak of the electrified body or bodies, the canal or the wire is understood not to be included unless it is specially mentioned.

Cavendish supposes the canal to be everywhere exactly saturated with the electric fluid, and that the only external force acting on the fluid in the canal is that due to the electrification of the other bodies.

Since this resultant force is not in general zero at all points of the canal, the fluid in the canal cannot be in equilibrium unless it is prevented from moving by some other force. Now the condition of incompressibility excludes any such displacement of the fluid as would alter the quantity of fluid in a given volume, and the stress by which such a displacement is resisted is called isotropic (or hydrostatic) pressure. In a hypothetical case like this it is best, for the sake of continuity, to suppose that negative as well as positive values of the pressure are admissible.

In the electrified bodies themselves the properties of the fluid are those defined in Art. 3. The fluid is therefore incapable of sustaining pressure except when its particles are close packed together, and as it cannot sustain a negative pressure, the pressure must be zero in the electrified bodies, and therefore also in the canal at the points where it meets these bodies.

The condition of equilibrium of the fluid in the canal is

P

dV dp
ds ds

+

where V denotes the potential of the electric forces due to the electrified bodies, p the density, and p the pressure of the fluid in the canal, and s the length of the canal reckoned from a fixed origin to the point under consideration.

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 A and A

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.

1

2

Let Vas before be the potential at any given point due to the electrified bodies, and let V, be its value in A,, and V, 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 V+V' 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 4, 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 ▲ 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 ma3, its potential, when b the distance between the disks is small compared with a the radius, will be approximately