Then we know that the potential of the globe at the end of the first part of the experiment cannot differ from zero by more than

1 d V, 486 D

where V is the potential of the shell when first charged.

But it appears from the mathematical theory that if the law of repulsion had been as (+), the potential of the globe when tested would have been by equation (25), p. 421,

0.1478 × qV.

T

Hence q cannot differ from zero by more than ±

1 d 72 D

Now, even in a rough experiment, D was certainly more than 300d. In fact no sensible value of d was ever observed. We may therefore conclude that q, the excess of the true index above 2, must either be zero, or must differ from zero by less than

Let the repulsion between two charges e and e' at a distance r be

where (r) denotes any function of the distance which vanishes at an infinite distance.

and f(r) is a function of r equal to fr [["$(r) dr] dr

(3)

(4)

We have in the first place to find the potential at a given point B due to a uniform spherical shell.

Let A be the centre of the shell, a its radius, a its whole charge, and

o its surface-density, then

a == Απασ.

(5)

Take A for the centre of spherical co-ordinates and AB for axis, and let AB = b.

Let P be a point on the sphere whose spherical co-ordinates are and, and let BP=r, then

and the potential due to the whole shell is therefore

(8)

We have next to determine the potentials of two concentric spherical shells, the radius of the outer shell being a and its charge a, and that of the inner shell being b and its charge B.

Calling the potential of the outer shell A, and that of the inner B, we find by what precedes,

In the first part of the experiment the shells communicate by the short wire and are both raised to the same potential, say V.

Putting _A = B = V and solving equations (16), (17), we find for the charge of the inner shell

B = 2Vb

bf (2a)-a{f(a + b) -ƒ (a - b)}

́ƒ (2a) ƒ (2b) – {ƒ (a + b) − ƒ (a−b)}* *

(18)

In the original experiment of Cavendish the hemispheres forming the outer shell were removed altogether from the globe and discharged. The potential of the inner shell or globe would then be

In the form of the experiment as repeated at the Cavendish Laboratory, the outer shell was left in its place, but was connected to earth, so that A = 0. In this case we find for the potential of the inner shell when tested by the electrometer

Let us now assume with Cavendish, that the law of force is some inverse power of the distance, not differing much from the inverse square, that is to say, let

If we suppose q to be a small numerical quantity, we may expand f(r) by the exponential theorem in the form

and if we neglect terms involving 9, equations (19) and (20) become

(23)

(24)

(25)

Laplace [Mec. Cel. 1. 2] gave the first direct demonstration that no function of the distance except the inverse square can satisfy the condition that a uniform spherical shell exerts no force on a particle within it.

If we suppose that B, the charge of the inner sphere, is always accurately zero, or, what comes to the same thing, if we suppose B1 or B, to be zero, then

9

bf (2a) - af (a + b) — af (a - b) = 0.

Differentiating twice with respect to b, a being constant, and dividing by a, we find

We may notice, however, that though the assumption of Cavendish, that the force varies as some inverse power of the distance, appears less general than that of Laplace, who supposes it to be any function of the distance, it is the most general assumption which makes the ratio of the force at two different distances a function of the ratio of those distances.

If the law of force is not a power of the distance, the ratio of the forces at two different distances is not a function of the ratio of the distances alone, but also of one or more linear parameters, the values of which if determined by experiment would be absolute physical constants, such as might be employed to give us an invariable standard of length.

Now although absolute physical constants occur in relation to all the properties of matter, it does not seem likely that we should be able to deduce a linear constant from the properties of anything so little like ordinary matter as electricity appears to be.

NOTE 20, ART. 272.

On the Electric Capacity of a Disk of sensible Thickness.

Consider two equal disks having the same axis, let the radius of either disk be a, and the distance between them b, and let b be small compared with a.

Let us begin by supposing that the distribution on each disk is the same as if the other were away, and let us calculate the potential energy of the system.

We shall use elliptical co-ordinates, such that the focal circle is the edge of the lower disk. In other words we define the position of a given point by its greatest and least distances from the edge of the lower disk, these distances being

a (a + B) and a (a-B).

The distance of the given point from the axis is

?= ααβ,

and its distance from the plane of the lower disk is

≈ = a (a2 − 1)3 (1 − ß2)a.

(1)

(2)

If A, is the charge of the lower disk, the potential at the given

1

point is

or, if we write

=

If A, is the charge of the upper disk, the density at any point is

2

(3)

(4)

(5)

(6)

(7)