where e is the charge of any element of the body, and the potential at that element.

The charge is

E=2(e).

(12)

Let us now suppose the electricity to become moveable and to distribute itself so as to be in equilibrium. The potential will then be uniform. Let its value be y, and since the charge remains the same the potential energy of the electrification in the state of equilibrium will be W1 = 14 E.

If K is the capacity of the conductor,

E=KY

(13)

(14)

(15)

Since W, the potential energy due to any arbitrary distribution of the charge, may be greater, but cannot be less than W, the energy of the same charge when in equilibrium, the capacity may be greater, but cannot be less, than

This inferior limit of the capacity is greater than that derived from the maximum value of the potential, and, as we shall see, sometimes gives a very close approximation to the true capacity.

In the case of the cylinder, if we suppose à to be uniform, and neglect the electrification of the flat ends

When the length of the cylinder is more than 100 times the diameter this value of the capacity is sufficiently exact for all practical purposes. The capacity of the inscribed ellipsoid is

To obtain a closer approximation let us suppose that the linear is expressed in the form A+ A, + &c. +A, the general term

density

If we consider a line of length 27 on which there is a distribution of electricity according to this law, and if ƒ and ƒ, are the distances of a given point from the ends of the line, and if we write

then the potential, Y, at the given point (a, ẞ), due to the distribution λ, is

n

72

where P is the same zonal harmonic as in equation (19), and Q, is the corresponding zonal harmonic of the second kiud*, and is of the form

In applying these results to the determination of the potential at any point of the axis of the cylinder we must remember that a point on the axis is at the distance b from any one of the generating lines of the cylinder, and therefore the potential at any point on the axis is the same as if the whole charge had been collected on one generating line. Hence at the point on the axis for which x = §, if we write

the potential due to the distribution whose linear density is

is

=AP

(24)

(25)

n(n−1)_n(n−1)(n−2) ̧n(n−1)(n−2) (n−3) _ &c.] (26)

L-n+

2.2

2.3.3

2.3.4.4

approximately, provided έ is between ± 1.

* See Ferrers' Spherical Harmonics, chap. v.

These values of the potential are calculated for the axis of the cylinder. The potential at the curved surface may be found from that at the axis by remembering that within the cylinder = 0. At a distance b from the axis the potential is therefore

where the values of 4 and its derivatives are those at the axis.

For a uniform distribution

(29)

which is approximately 24 when έ=0, and

Hence, when the length of the cylinder is many times its diameter, the potential at the axis may be taken for that at the surface in approximations of the kind here made.

We have next to find the integral of the product of the density into the potential. We may consider the product of each pair of terms by itself. If we write 2 for the value of L when έ= l, or approximately

Determining A, so as to make Л(^,+λ ̧)(4 ̧+41⁄2)dæ a minimum, we find

2

This approximation is evidently of little use unless the length of the cylinder considerably exceeds 7.245 times its diameter, for this ratio makes the second term of the denominator infinite. It shows, however, that when the ratio of the length to the diameter is very great, the true capacity approximates to the value of K, given in (18).

We may proceed in the same way to determine A, and A, so that S(λ。 + d ̧ + d ̧) (4% + 42 + 41) dx

2

2

shall be a minimum, and we thus find a third approximation to the value of the capacity, in which

so that when is very large the distribution approximates to

The value of the inferior limit of the capacity, as given by this approximation is

As & increases, K approaches to the value found by the first approximation.

To indicate the degree of approximation, the value of £ and of the successive terms of the denominator are given below.

The observed capacities of Cavendish's cylinders may be deduced from the numbers given in Art. 281 by taking the capacity of the globe of 12.1 inches diameter equal to 6-05, and their capacities as calculated by the formula of this note are given in the following table.

The agreement of the calculated and measured values is remarkable.

NOTE 13, ARTS. 152, 280.

Two cylinders.

In the case of two equal and parallel cylinders at distance c, the linear densities being uniform and equal to A, and A,, the part of the potential energy arising from their mutual action is

If the two cylinders are in electric communication with each other λ1 =λ, and the capacity of the two cylinders together is approximately

If a cylinder is placed at a distance d from a conducting plane surface and parallel to it, then the electric image of the cylinder will be at a distance c = 2d, and its charge will be negative, so that the capacity