The whole of the positive fluid is thus condensed on the outer surface, the whole of the negative fluid distributed within the inner sphere, and the shell between the two spherical surfaces is entirely deprived of both fluids.

At the outer surface, the force on the positive fluid is from the centre, but the fluid there cannot move, because it is prevented by the insulating medium which surrounds the sphere.

In the shell between the two spherical surfaces the force on the positive fluid would be from the centre. Hence if any positive fluid enters this shell, it will be driven to the outer surface, and if any negative fluid enters, it will be driven to the inner surface.

But all the positive fluid is already at the outer surface, and all the negative fluid is already in the inner sphere, where, as Green has shown, it is in equilibrium, and thus the fluids are in equilibrium throughout the sphere.

It may be remarked that this solution, according to which a certain portion of matter becomes entirely deprived of both fluids, is inconsistent with the ordinary statements of the theory of two fluids, which usually assert that bodies, under all circumstances, contain immense quantities of both fluids.

In the two-fluid theory, by depriving matter of both fluids, we get an inactive substance which gives us no trouble, but in the one-fluid theory, matter deprived of fluid exerts a strong attraction on the fluid, the consideration of which would considerably complicate the mathematical problem.

Infinite plate with plane parallel surfaces.

The distribution of the fluid in an infinite plate with plane parallel surfaces is given in the general solution which we have obtained for a body bounded by a quadric surface, namely, p = Cp”~*.

In the case of the plate we must suppose it bounded by the planes x = + α, and x = -a, and then p is defined by the equation

If σ is the quantity of fluid in a portion of the plate whose area is unity,

The distribution in an infinitely thin disk may be deduced from that in an ellipsoid by making one of the axes infinitely small. It is better

DISTRIBUTION OF FLUID ON A ROD.

373

however to proceed by the method which we have already employed, only that instead of supposing the line A,PA, (Fig. p. 368) to be a double cone, we suppose it to be a double sector cut from the disk. The breadth of this sector is proportional to the distance from P, so that the condition of equilibrium of the repulsions of two corresponding elements whose surface-densities are σι and is

3-n

2 2 2

whence we find, as before, that if the equation of the edge of the disk is

The quantity of fluid in the disk is found by integrating over the surface of the disk, and is

Hence if σ is the mean surface-density, the surface-density at any point is given by the equation

The distribution on an infinitely thin rod is found by considering APA, a rod of uniform section, which leads to the equation

where A is the linear density, and if the length of the rod is 2a, and if x is the distance from the middle, and x2 = a2 (1 − p3), the distribution of the linear density is given by

when n = 2, AA, 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” is

We thus find for the potential of the sphere

1

n-1

When n becomes equal to 4, V becomes infinite.

When n is equal to 2, V = Qa ̄1.

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.

CANALS OF INCOMPRESSIBLE FLUID.

375

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

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 = p, = 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

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 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