as in the case of the sphere, but greater at the sharp ends of the major axis than at the equator, and the ratio of the densities increases indefinitely as we make the ellipsoid sharper and sharper. This leads us to state a principle of great importance in the theory of electrical distribution, viz., that the electrical density is very great at any pointed part of a conductor.

If we determine the ratio of the densities at two points of an ellipsoid1, diminish the charge, and redetermine the same ratio, we shall find that, although the actual densities are diminished, the ratio remains the same; and if we determine the density at any point of the ellipsoid, and then halve its charge by touching it with an equal and similar ellipsoid (they must be placed with their axes in the same straight line, and made to touch at the poles), and redetermine the density at the same point as before, we shall find that the density in the second case is half that in the first. We have in fact, in general, the important proposition that—

The density at any point of a conductor is proportional to the whole charge on the conductor, or, what is the same, to the mean density.

The following case given by Coulomb is interesting; it shows the tendency of electricity towards the projecting parts, ends, or points of bodies. The conductor was a cylinder with hemispherical ends,-the length of the cylinder being 30 inches, its diameter 2 inches. Coulomb gives the following results:

1 38-6

greater than for the larger. The above result also affords an experimental illustration of the action of the earth in discharging a conductor connected with it. Comparing the conductor to the small sphere and the earth to the large sphere of 62 times the superficial area of the small one, if we start with charge Q on small sphere, and then put the two in contact, the charge on the small sphere will be reduced to- Q, so that the mean density is diminished in the ratio 1 : 386. This ratio increases indefinitely as the ratio increases. These results are in satisfactory agreement with Poisson's calculations. Coulomb was led by his observations to assign 2 as the limit of the ratio of the mean densities when the ratio of the diameters of the spheres is infinitely great; the mathematical theory gives

S'

S

Coulomb also determined the density at the apex or smaller end of the body formed by two unequal spheres in contact. The following are his results, the mean density of the larger sphere being unity :

the middle.

Other results, taken from Coulomb's unpublished papers, may be found in Biot, Mascart, or Riess. His results for a circular disc we shall quote further on.

Riess made a series of experiments on cubes, cones, etc.; but as these are not of theoretical interest, the calculation in such cases being beyond the powers of analysis at present, and as the use of the proof-plane or sphere with bodies where edges and points occur is not free from objection, we content ourselves with referring to Riess's

work for an account of the results.

tors.

Coulomb made a series of experiments on Coulomb's researches bodies of different forms, which he built up out on compo- of spheres of different sizes, or out of spheres site conduc- and cylinders. These are of very great interest, partly on account of the close agreement of some of the results with the deductions subsequently made by Poisson from the mathematical theory, and partly on account of the clearness with which they convey to the mind the general principles of electric distribution. His method in most cases was to build up the conductor and electrify it with all the different parts in contact, and then, after separating the parts widely, to determine the mean density or the whole amount of electricity on each part by the proof-plane or otherwise.

When two equal spheres are placed in contact the distribution will of course be the same in each; Coulomb found that, from the point of contact up to a point on the surface of either sphere distant from it by about 20°, no trace of electricity could be observed; at 30°, 60°, 90°, values 20, 77, 96, 1.00. When the spheres are unequal, 180° respectively, the electric density had the relative the distribution is no longer alike on each. On the small sphere it is less uniform, and the density at the point of the small sphere diametrically opposite the point of contact is greater than anywhere else on the body. The distribution on the larger sphere is more uniform than on the smaller, and the more unequal the spheres are the more uniform is the distribution on the larger, and the smaller the unelectrified part in the neighborhood of the point of contact. The following results of Coulomb are useful illustrations of distribution on elongated and pointed bodies:

Three equal spheres (2 in. diameter) in contact, with their centres in the same straight line: the mean densities were 1.34, 1.00, 1.34 on the spheres 1, 2, and 3 respectively. 1-56, 105, 1.00. Six equal spheres as before: mean densities on 1, 2, and 3

Twelve equal spheres: mean densities on 1, 2, and 6 — 1.70, 1.14, 1.00.

Twenty-four equal spheres: mean densities on 1, 2, and 12 — 1.75, 1.07, 1:00.

Large (8 in. diameter) sphere with four small (2 in.) spheres applied to it, all the centres in line: the mean density on large sphere being 1, that on the small one next it was 60, that on

the extreme small one 2:08.

From this it appears that although the whole amount of electricity on the large sphere is greater than that on the small, yet the mean density for the smaller sphere is

1 We suppose in all these experiments that we are dealing with a single body, sufficiently distant not only from all electrified bodies but from all neutral conductors to be undisturbed by them. This condition is essential.

It would not do to make the pole of one touch the equator of the other, or to place them otherwise unsymmetrically. 2 Traité de Physique.

MATHEMATICAL THEORY OF ELECTRICAL

EQUILIBRIUM.

We take as the basis of our theory the assumptions already laid down under the head Provisional Theory, and in addition the precise elementary law of electrical action established by Coulomb. We shall also suppose that we have only perfect conductors and perfect non-conductors to deal with, the medium being in all cases the same, viz., air. When we have to deal with electrified non-conductors we shall suppose the electrification to be rigid, i. e., incapable of disturbance by any electric force we have to consider.

In our mathematical outline we have in view the requirements of the physical more than the mathematical student, and shall pass over many points of great interest and importance to the latter, for full treatment of which we must refer him to original sources, such as the classical papers of Green, the papers of Sir William Thomson, and the works of Gauss. Of

peculiar interest mathematically is the elegant and powerful memoir of the last-Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungskräfte, in which will be found detailed discussions of the continuity of the integrals used in the potential theory, etc. The works of Green and Thomson are too well known in this country to require farther

remark.

Definitions,

When, in what follows, we speak of the electric field, we mean simply a portion of space which we are considering with reference to its electrical properties; it will be found conducive to clearness to regard that space as bounded. In general the natural boundary would be the walls of the experimenting room; but, for mathematical purposes, we shall, unless the contrary is stated, suppose our field to be bounded by a sphere of radius so great that the action at a point on its circumference due to an electrified body in the field is infinitely

small.

The resultant force at a point in the electric field is the force which would be exerted on a unit of electricity placed there without disturbing the electrical distribution elsewhere. It is plain that the resultant force has a definite magnitude and direction at every point in the field, and consequently is in modern mathematical language a vector. A curve drawn in the field such that its tangent at every point is in the direction of the resultant force at that point is called a line of force. We can draw such a line through every point of space, and if we suspend at any point a small conducting needle, it is obvious, from what we have already laid down about induction, that it will take up a position very nearly parallel to the line of force; so that if we start from any point and carry the centre of the needle always in the direction in which the needle points, we should trace out a line of force.

The potential at any point is the work done by a unit of electricity in passing from that point to the infinitely distant boundary of the electric field, the electric distribution being supposed undisturbed. It is usual to call the infinitely distant boundary a place of zero potential. Zero is to be understood in the sense of "point or position from which we reckon." 1

Force in

Consider two points P, Q, infinitely near each terms of v. other in the field, and draw a curve from P passing through Q to co. Then, if F be the component parallel to PQ of the resultant force at P, we have by our definition

dv dx

dV dz

(3)

dv dy We may remark that, in all cases which we shall consider at present, the work done in passing from any point to any other point is the same whatever the intermediate path of our exploring unit. Hence V as above defined is a single valued function, and the formula (3) gives the components of resultant force when V is known.

The work done by a unit of + electricity in passing by any path from P to Q is called the electro-motive force from P to Q; it is obviously equal to the difference of the potentials at the two points. Thus

Vp_\

where D denotes +(-)2 + (n − y)3 + (5 − x)2, and the integral is to be extended over every part of the field where there field, on the understanding that p= 0 where there is no charge. is any charge,-or, which is the same thing, over the whole

If, as will generally be the case, the electricity is distributed on a surface in such a way that on an element dS of surface there is a quantity odS of electricity, where a is a finite surface density, then

where D has the same meaning as before, and the integral is extended all over the electrified surface or surfaces.

We may make here the important remark that, Continuity and (8) are finite and continuous. This depends so long as p or σ is not infinite, the integrals in (7) of V. on the fact, which we cannot stop to prove, that the part of the potential at P, contributed by an infinitely small portion of electricity surrounding P, is infinitely small.

occur in nature.

In practice, therefore, the electric potential is always continuous; for although we may in theory speak of discrete points and electrified lines where finite electrification is condensed into infinitely small space, yet no such cases ever It may also be shown for any electrical system of finite extent, that, as the distance of P from O, any fixed point at a finite distance from the system is increased indefinitely, the potential at P approaches more and M more nearly the value where M is the algebraical sum of all the electricity in the system, and D the distance of P from O. Hence at any point infinitely distant from O, V = 0.

D'

Surface in

tegral of electric in

duction.

We next proceed to prove the following proposition, which will form the basis of the subsequent theory :The surface integral of electric induction taken all over the surface inclosing any space is equal to 4 times the algebraical sum of all the electricity in that space. By the electric induction across any element of the surface (taken so small that the resultant force at every point of it may be regarded as uniform) is meant the product of the area of the element into the component of the resultant force in the direction of the normal to the element which is drawn outwards with respect to the inclosed space. Thus dS being an element of surface, e the angle between the positive direction of the resultant force R and the outward normal, and E the sum of all the electricity in the inclosed space, the proposition in symbols is

We shall prove it in the manner most naturally suggested by the theory of electrical elements acting at a distance, by first showing that it is true for a single element e either outside or inside the surface. Let us suppose e to be at a point P, fig. 11, within S, which for greater generality we may suppose to be a re-entrant surface. Draw a small cone of vertical solid angle do at P, and let it cut the surface in the elements QR, Q'R', Q"R"; let the outward normals to these be QM, Q'M', Q"M". The elements of the surface integral contributed by QR, Q'R', QRe cose

Combining these results, we see that the proposition is true for ■ single element. Hence, by summation for all the elements, we can at once extend it to any electrical system; for all the elements external to S give zero, and all the internal elements give 4x20- 4 E. Let us apply the above proposition to the space Equation enclosed by the infinitely small parallelepiped of Laplace whose centre is at xyz, and the co-ordinates of and Poisson. whose angles are xdx, y±jdy, z±jdz. The contributions to the surface integral from the two dv dx day faces perpendicular to the x-axis are — + dydz dx 2 dx2 (dv dx d2V dyda. Adding these and the four parts dx 2 dx2 from the remaining sides, and equating to 4wpdxdydz, which is the 4E in this case, we have

and

ᏯᏙ ᏯᏙ ᏯᏙ

+ + + 4xp=0, dx dy da

or, as it is usually abbreviated,

(10) Equation (10), originally found by Laplace for the case p = 0, and extended by Poisson, has been called the characteristic equation of the potential. It may be applied at any point where P is finite and the electric force continuous. It might be shown by examining the integrals representing X, Y, Z, and etc., that the electric force is continuous wherever there is finite volume density. Equation (10) may be looked on either as an equation to determine the potential when p is given, or as an equation to determine when V is given. We shall have

dv

dx

occasion to use it in both ways.

Conditions at an elec

trified surface.

P

The characteristic equation cannot be applied in the form (10) when the resultant force is discontinuous. This will be found to be the case at a surface on which electricity is distributed with finite surface density. Let us consider the values of the resultant force at two points, P and Q, infinitely near each other, but on opposite sides of such a surface. Resolve the resultant force tangentially and normally to the surface. If we consider the part of the force which arises from an infinitely small circular disc, whose radius, though infinitely small, is yet infinitely great compared with the distance between P and Q, we see that infinitely little is contributed to the tangential component at P or Q by this disc, while it can be readily shown that the part of the normal component arising therefrom is 20, directed from the disc in each case, when o is the surface density. Hence, since the part of the resultant force arising from all the rest of the electrified system obviously is not discontinuous between P and Q, the tangential component is continuous when we pass through an electrified surface, but the normal component is suddenly altered by 4ño.

For a thorough investigation of these points the reader is referred to Gauss or Green. We can arrive very readily at the amount of the discontinuity of the normal force by applying (9) to the cylinder formed by carrying an infinitely short generating line round the element dS, so that one end of the cylinder is on one side of dS and the other on the other, the lateral dimensions being infinitely small, but still infinitely greater than the longitudinal. The only part of the integral which is of the order of dS is the part arising from the two ends; hence if N, N' be the value of the normal components on the two sides of S, we clearly get

(N- N') dS= 4nodS, or N-N' — 4′′σ.

If Vi, V, denote the potentials on the two sides of S, and v1, the normals to ds, drawn towards these sides respectively,

then we may obviously write our equation

We have seen that we can draw through every Level point of the electric field a line of force. surface, surface constructed so that the potential at every point of it has the same value is called an equipotential or level surface. We can obviously draw such a surface passing through every point of the field. It is clear, too, that the line of force at every point must be perpendicular to the level surface passing through that point. For since no work is done level surface to a neighboring point, there can be no comon a unit of electricity in passing from one point of a ponent of the resultant force tangential to the surface; in other words, the direction of the resultant force is perpendicular to the surface. This is expressed otherwise by saying that the lines of force are orthogonal trajectories to the level

surface.

Tubes of

If we take a small portion of a level surface, and draw through every point of the boundary force. surface which will cut orthogonally every level surface a line of force, we shall thus generate a tubular which it meets. Such a surface is called a tube of force.

Let a tube of force cut two level surfaces in the elements dS and dS'. Apply to the space contained by the part of the tube between the surfaces our fundamental equation (9). We thus get, since there is no normal component perpendicular to the generating lines of the tube,

provided the tube does not cut through electrified matter between the two surfaces. Here R and R' denote the resultant force at dS and d'S, which are supposed so small that the force may be considered uniform all over each of them. It appears then that the product of the resultant force into the area of the normal section of a tube of force is constant for the same tube so long as it does not cut through electrified matter; or, what amounts to the same, the resultant force at any point of a tube of force varies inversely as the normal section of the tube at that point.

Important tubes of property of force RdS -R'dS'.

If we divide up any level surface into a series of small elements, such that the product RdS is constant for each element and equal to unity, and draw tubes of force through each small element, then the electric induction through any finite area of the surface is equal to the number of tubes of force which pass through that area; for if n be that number, we have, summing over the whole of the area— ΣRdS=n,

. . (13) the left-hand side of which is the electric induction through the finite area. It is clear, from the constancy of the product RdS for each tube of force, that if this is true for one level surface it will be true for every other cut by the tubes of force. It is evident that the proposition is true for any surface, whether a level surface or not, as may be seen by projecting the area considered by lines of force on a level surface, and applying to the cylinder thus formed the surface integral of electric induction, it being remarked as obvious that the same number of tubes of force pass through the area as through the projection. This enables us to state the proposition involved in equation (9) in the following manner:

The excess of the number of tubes of forces which Charge leave a closed surface over the number which enter measured is equal to 4 times the algebraical sum of all the by tubes of force. electricity within the surface.

(N.B.-The positive direction of a line of force is that direction in which a unit of electricity would tend to move along it.) This proposition enables us to measure the charge of a body by means of the lines' of force. We have only to draw a surface inclosing the body, and very near to it, and count the lines of force entering and leaving the surface. If the number of the latter, diminished by the

1 Here we drop the distinction between line and tube of force. We shall hereafter suppose the lines of force to be always drawn so as to form unit tubes, and shall speak of these tubes as lines of force, thereby following the usual custom.