If we consider a line of length 27 on which there is a distribution of electricity according to this law, and if f, and f, 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

=AQ (a) P(B),

(21)

where P is the same zonal harmonic as in equation (19), and Q1 is the corresponding zonal harmonic of the second kiud*, and is of the form

Q (a) vanishes when a is infinite. The values of the first four harmonics of the second kind are

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=έ, if we write

the potential due to the distribution whose linear density is

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 and its derivatives are those at the axis.

For a uniform distribution

(29)

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 £ for the value of L when = 1, or approximately

Determining A, so as to make Л(+λ ̧)(4+4)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 4, so that

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 A,A,, 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

of the cylinder will be increased. The capacity of the cylinder in presence of a conducting plane at distance c, is

Thus in Cavendish's experiment he used a brass wire 72 inches long and 0.185 in diameter. The capacity of this wire at a great distance from any other body would be 5.668 inches. Cavendish placed it horizontally 50 inches from the floor. The inductive action of the floor would increase its capacity to 5.994 inches; Cavendish, by comparison with his globe, makes it 5-844.

To compare with this he had two wires each 36 inches long and 0.1 inch diameter.

The capacity of one of these at a distance from any other body would be 2.8697 inches, or the two together would be 5·7394 inches.

The two wires were placed parallel and horizontal at 50 inches from the floor. Each wire was therefore influenced by the other wire, and also by the negative images of itself and the other wire.

The denominator of the fraction expressing the capacity is therefore

The numerator of the fraction which expresses the capacity of both wires together is 36, so that the capacity of the two is

If we suppose the plate AB to be overcharged and the plate DF to be equally undercharged, the redundant fluid in any element of AB being numerically equal to the deficient fluid in the corresponding element of DF, then what Cavendish calls the repulsion on the column CE in opposite directions becomes in modern language the excess of the potential at C over that at E. Hence the object of the Lemma is to determine approximately the difference of the potentials of two curved plates when their equal and opposite charges are given, and to deduce their charges when the difference of their potentials is given.