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

¿ (log 7 – 1).

NOTE 5, ART. 90.

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.

66

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

cube, and the force between the two molecules must vary as the total force between the sets of molecules separated by the imaginary plane, and therefore the product of the repulsion between two given molecules into the distance between them must be constant, in other words the repulsion varies inversely as the distance.

In this demonstration the repulsion considered is that between two given molecules, and it is shown that this must vary inversely as the distance between them in order to account for Boyle's law of the elasticity of air.

If, however, we suppose the same law of repulsion to hold for every pair of molecules, Newton shows in his Scholium that it would require a greater pressure to produce the same density in a larger mass of air.

We must therefore suppose that the repulsion exists, not between every pair of molecules, but only between each molecule and a certain definite number of other molecules, which we may suppose to be defined as those nearest to the given molecules. Newton gives as an example of such a kind of action the attraction of a magnet, the field of which is contracted when a plate of iron is interposed, so that the attractive power appears to be bounded by the nearest body attracted.

If the repulsion were confined to those molecules which are within a certain distance of each other, then, as Cavendish points out, the pressure arising from this repulsion would vary nearly as the square of the density, provided a large number of molecules are within this distance. Hence this hypothesis will not explain the fact that the pressure varies as the density.

On the other hand, if the repulsion were limited to particular pairs of particles, then since the particles are free to move, these pairs of particles would move away from each other till only those particles were near each other between which the repulsive force is supposed not to exist.

It would appear therefore that the hypothesis stated by Newton and adopted by Cavendish is the only admissible one, namely, that the repulsive force is inversely as the distance, but is exerted only between the nearest molecules.

Newton's own conclusion to his investigation of the properties of air on the statical molecular hypothesis is as follows:-"An vero Fluida Elastica ex particulis se mutuo fugantibus constent, Quæstio Physica est. Nos proprietatem Fluidorum ex ejusmodi particulis constantium mathematice demonstravimus, ut Philosophis ansam præbeamus Quæstionem illam tractandi."

The theory that the molecules of elastic fluids are in motion satisfies the conditions of the question as pointed out by Newton in a much more natural manner than any modification of the statical hypothesis.

According to the kinetic theory of gases, each molecule is in motion, and this motion is during the greater part of its course undisturbed by

the action of other molecules, and is therefore uniform and in a straight line. When however it comes very near another molecule, the two molecules act on each other for a very short time, the courses of both are changed and they go on in the new courses till they encounter other molecules.

It would appear from the observed properties of gases that the mutual action between two molecules is insensible at all sensible distances. As the molecules approach, the action is at first attractive, but soon changes to a repulsive force of far greater magnitude, so that the general character of the encounter depends mainly on the repulsive

force.

On this theory, the elasticity of the gas may still be said in a certain sense to arise from the repulsive force between its molecules, only instead of this repulsive force being in constant action, it is called into play only during the encounters between two molecules. The intensity of the impulse is not the same for all encounters, but as it does not depend on the interval between the encounters, we may consider its mean value as constant. The average value of the force between two molecules is in this case the value of the impulse divided by the time between two encounters. Hence we may say that the force is inversely as the distance between the molecules, and that it acts between those molecules only which encounter each other.

For an earlier investigation by Cavendish of the properties of an elastic fluid, see Note 18.

NOTE 7, ART. 101.

Here Cavendish endeavours to fix a precise meaning to the terms "positively and negatively electrified," terms which he found current among electricians, but not well defined. The meaning which he here fixes to them, and which he afterwards makes much use of, is equivalent to the meaning of the modern term potential, as used by practical electricians. The idea of potential as used by mathematicians is expressed by Cavendish in his theory of canals of incompressible fluid.

In the "Thoughts concerning Electricity," and in the unpublished papers, degrees of electrification are spoken of. These degrees of electrification are measured in the experimental researches by means of electrometers of different kinds, and since he has compared the indications of his electrometers with the degrees of electrification required to make a spark pass between the balls of Lane's discharging electrometer, we may express all these measurements in modern units, though Cavendish's original electrometers no longer exist.

I have not been able to trace the idea of electric potential in the work of Epinus, so that Cavendish seems the first to have made use of it. The relation between the charge of a body and the degree of

its electrification is the main object of Cavendish's experimental researches, and the results of his work were expressed in the material form of a collection of coated plates, each of which had a capacity equal to that of a sphere of known diameter.

The leading idea in the great experimental work of Coulomb seems to be the measurement of the charges of the different bodies of a system and of parts of these bodies. Perhaps the most valuable of Coulomb's many contributions to experimental physics was the measurement of the surface-density of the distribution of electricity on a conductor on different parts of its surface by means of the proof plane. The numerical results obtained by Coulomb led directly to the great mathematical work of Poisson. I have not been able, however, to trace, even in those parts of Coulomb's papers where it would greatly facilitate the exposition, any idea of potential as a quantity which has the same value for all parts of a system of conductors communicating with each other.

NOTE 8, p. 51.

Cases of Attraction and Repulsion.

The statements of Cavendish may be illustrated by the case of two spheres A and B, whose radii are a and b, and the distance between their centres c.

If the charge of A is 1, and that of B is 0, the attraction is

an expression which shows that it depends chiefly on the value of b, the radius of the sphere without charge.

If the sphere B, instead of being without charge, is at potential zero, that is, if it is not insulated, the attraction is

The number of times that the attraction of an uninsulated sphere exceeds that of a sphere without charge is therefore approximately

which is greater as the sphere is smaller. This agrees with what Cavendish says in Art. 108.