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With respect to two bodies at the same potential, Cavendish remarks in Art. 113, that it may be said that one of them may be rendered undercharged in the part nearest to the other, and he shows that even in this case, the two bodies must repel each other. But it may be shown that each of the bodies must be overcharged in every part of its surface. For in the first place no part can be undercharged, for the lines of force which terminate in an undercharged surface must have come from an overcharged surface at which the potential is higher than at the surface. But there is no body in the field at a higher potential than the two bodies considered. Hence no part of their surface can be undercharged. Nor can any finite part of the surface be free from charge, for it may be shown that if a finite portion of the surface of a conductor is free from charge, every point which can be reached by continuous motion from that part of the surface without passing through an electrified surface must be at the same potential. Hence no fiuite portion of a surface can be free from charge, unless the whole surface is free from charge.

NOTE 9, ART. 124.

The rate at which electricity passes from a conductor to the surrounding air or from the surrounding air to a conductor was believed to be much greater by Cavendish and his contemporaries than is consistent with modern experiments. Judging from the statements of the electricians of each generation, it would seem as if this rate had been diminishing steadily during the last hundred years in exact correspondence with the improvements which have been made in the construction of solid insulating supports for electrified conductors.

Whenever the intensity of the electromotive force at the surface of a conductor is sufficiently great, the air no doubt becomes charged.* This is the case at a sharp point connected with the conductor even when the potential is low, but when the curvature of the surface is continuous and gentle, the conductor must be raised to a high potential before any discharge to air begins to take place.

Thus in Thomson's portable electrometer, in which there are two disks placed parallel to each other at different potentials, the percentage loss of electricity from day to day is very small, and seems to depend principally on the solid insulators, for when the disks are placed very near each other, less loss is observed than when they are further apart, though the intensity of the force urging the electricity through the intervening stratum of air is greater the nearer the disks are to each other.

* M. R. Nahrwold (Wiedemann's Annalen v. (1878) p. 440) finds that the discharge from a sharp point communicates a charge to dusty air which can be detected in the air for some time afterwards. This does not occur in air free from dust. But the discharge from an incandescent platinum wire communicates a lasting charge even to air free from dust.

On the surface density of electricity near the vertex of a cone.

Green has given in a note to his Essay, section (12), the following results of an investigation which, so far as I am aware, he never published.

"Since this was written, I have obtained formulæ serving to express, generally, the law of the distribution of the electric fluid near the apex 0 of a cone, which forms part of a conducting surface of revolution having the same axis. From these formulæ it results that, when the арех of the cone is directed inwards, the density of the fluid at any point p, near to it, is proportional to; r being the distance Op, and the exponent n very nearly such as would satisfy the simple equation

(4n+2) B = 3π,

where 2ẞ is the angle at the summit of the cone.

If 2ẞ exceeds 7, this summit is directed outwards, and when the excess is not very considerable, n will be given as above: but 2ẞ still increasing, until it becomes 2 - 2y, the angle 2y at the summit of the cone, which is now directed outwards, being very small, n will be given by

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and in case the conducting body is a sphere whose radius is b, on which P represents the mean density of the electric fluid; p, the value of the density near the apex 0, will be determined by the formula

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Professor F. G. Mehler* of Elbing has investigated the distribution of electricity on a cone under the influence of a charged point on the axis, and the inverse problem of the distribution on a spindle formed by the revolution of the segment of a circle about its chord.

He finds that when the segment is a very small portion of the circle, so that the conical points of the spindle are very acute, the surfacedensity at any point is inversely proportional to the product of the distances of that point from the two conical points.

Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function, und ihre Anwendung in der Theorie der Electricitätsvertheilung. (Elbing, 1870.)

NOTE 10, p. 63.

Sir W. Thomson* has determined in absolute measure the electromotive force required to produce a spark in air between two electrodes in the form of disks, one of which was plane, and the other slightly convex, placed at different distances from each other. Mr Macfarlanet has recently made a more extensive series of experiments on the disruptive discharge of electricity. He finds that in air at the ordinary pressure and temperature the electromotive force required to produce a spark between disks, 10 cm. diameter, and from 1 to 0·025 cm. apart, is expressed by the empirical equation

V = 66·940 (s2+205038),

where s is the distance between the disks.

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If we suppose that in the space between the disks the potential varies uniformly, as it does between two infinite planes, then the resultant electromotive intensity is R

=

V

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If, on the other hand, we suppose that the variation of the potential near the surface of the disks is affected by unknown causes, we would get a better estimate of the intensity by taking

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Both

dV
ds

and diminish as the distance increases, approximating to the limit 66.940.

This corresponds to a surface-density of 5-327 units of electricity per square centimetre, and to a tension of 178.3 dynes per square centimetre. As the ordinary pressure of the atmosphere is about a million dynes per square centimetre, the pressure with which the electricity tends to break through the air is only about pressure of the atmosphere.

1

5600

of the

If the electrodes are convex surfaces, whose radii of curvature, a and b, are large compared with the least distance c between the surfaces, then if

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the greatest electric force at the surface whose radius is a will be equal to that at either of two parallel plane surfaces at the same potentials whose distance is s.

* Proc. R. S., 1860, or Papers on Electrostatics, chap. xIx.
+ Trans. R. S. Edin., Vol. xxvIII., Part II. (1878), p. 633.

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Hence the electromotive force required to produce a spark between convex surfaces, as in Lane's electrometer, is less than if the surfaces had been plane and at the same distance.

When the air-space is large, the path of the sparks, and therefore the electromotive force required to produce them, is exceedingly irregular. The accompanying figure is from a photograph of a succession of sparks taken between the same electrodes from four Leyden jars charged by Holtz's machine.

A portion of the path near the positive electrode is nearly straight, there is then a sharp turn, which, in all the sparks represented, is in the same direction. Beyond this the course of the spark is very irregular, although its general direction is deflected towards the same side as the first sharp turn.

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Theory of two circular disks on the same axis, their radii being small compared with the distance between them.

A circular disk may be considered as an ellipsoid, two of whose axes are equal, while the third is zero, and we may apply the method of ellipsoidal co-ordinates to the calculation of the potential*. In the

* See Ferrers' Spherical Harmonics, p. 136.

case before us everything is symmetrical about the axis, so that we have to consider only the zonal harmonics, and of these only those of even order, unless we wish to distinguish between the surface density on opposite sides of the same element of the disk, for this depends on the harmonics of odd orders.

Let a be the radius of the first disk, b that of the second, and c the distance between them.

We shall use ellipsoidal co-ordinates confocal with the first disk. Let r, and r, be the greatest and least distances respectively of a given point from the edge of the disk, and let

a3 − } (r ̧ − r ̧)2 = a3μ3,

(1)

} (r, + r ̧)2 — a2 = a2v3,

(2)

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then if is the distance of the point from the plane of the disk, and r its distance from the axis,

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If the surface-density of the electricity on the disk is a function of the distance from the axis, it may be expressed in the form

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and P is the zonal harmonic of order 2n. Only even orders are admissible, for since every element of the disk corresponds to two values of μ, numerically equal but of opposite signs, a term involving an harmonic of odd order would give the surface-density everywhere zero.

The potential arising from this distribution at any point whose ellipsoidal co-ordinates are w = αμ and ηπι = av

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2n

In this expression Q'(v) denotes a series, the terms of which are numerically equally to those of Q2, (v), the zonal harmonic of the second kind, but with the second and all even terms negative. If we put i for 1, we may write

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This expression is an infinite series, the terms of which increase without limit when y is diminished without limit.

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