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In describing Thomson's early electrical researches we shall not enter into detailed calculations, but merely explain the methods employed. The meaning of certain technical terms may be recalled in the first place.

The whole space in which a distribution of electricity produces any action on electrified bodies is called the electrical field of the distribution. The force exerted on a very small insulated trial conductor, on which is an electric charge of amount equal to that taken as the unit quantity of electricity, measures the field-intensity at any point at which the conductor is placed. The direction of the field-intensity at the point is that in which the small conductor is there urged. If the charge on the small conductor were a negative unit, instead of a positive, the direction of the force would be reversed; the magnitude of the force would remain the same. To make the field-intensity quite definite, a positive unit is chosen for its specification. For a charge on the trial-conductor consisting of any number of units, the force is that number of times the field-intensity. The field-intensity is often specified by its components, X, Y, Z in three chosen directions at right angles to one another.

Now in all cases in which the action, whether attraction or repulsion, between two unit quantities of matter concentrated at points is inversely as the square of the distance between the charges, the field-intensity, or its components, can be found from a certain function V of the charges forming the acting distribution [which is always capable of being regarded for mathematical purposes as a system of small charges existing at points of space, point-charges we shall call them], their positions, and the position of the point at which the field-intensity is to be found. If q₁, q₂, ... be the point-charges, and be positive when the charges are positive and negative when the charges are negative, and r₁, r₂, ... be their distances from the point P, V is q₁ ⁄ r₁ + q₂ ⁄ r₂ + ... The field-intensity is the rate of diminution of the value of V at P, taken along the specified direction. The three gradients parallel to the three chosen coordinate directions are X, Y, Z; but for their calculation it is necessary to insert the values of r₁, r₂, ... in terms of the coordinates which specify the positions of the point-charges, and the coordinates x, y, z which specify the position of P. Once this is done, X, Y, Z are obtained by a simple systematic process of calculation, namely, differentiation of the function V with respect to x, y, z.

This function V seems to have been first used by Laplace for gravitational matter in the Mécanique Céleste; its importance for electricity and magnetism was recognised by Green, who named it the potential. It has an important physical signification. It represents the work which would have to be done to bring a unit of positive electricity, against the electrical repulsion of the distribution, up to the point P from a point at an infinite distance from every part of the distribution; or, in other words, what we now call the potential energy of a charge q situated at P is qV. The excess of the potential at P, over the potential at any other point Q in the field, is the work which must be spent in carrying a positive unit from Q to P against electrical repulsion. Of course, if the force to be overcome from Q to P is on the whole an attraction, work has not been spent in effecting the transference, but gained by allowing it to take place. The difference of potential is then negative, that is, the potential of Q is higher than that of P.

The difference of potential depends only on the points P and Q, and not at all on the path pursued between them. Thus, if a unit of electricity be carried from P to Q by any path, and back by any other, no work is done on the whole by the agent carrying the unit. This simple fact precludes the possibility of obtaining a so-called perpetual motion (a self-acting machine doing useful work) by means of electrical action. The same thing is true mutatis mutandis of gravitational action.

In the thermal analogy explained by Thomson in his first paper, the positive point-charges are point-sources of heat, which is there poured at constant rate into the medium (supposed of uniform quality) to be drawn off in part from the medium at constant rate where there are sinks (or negative sources),—the negative point-charges in the electrical case,—while the remainder is conducted away to more and more distant parts of the conducting medium supposed infinitely extended. Whenever a point-source, or a point-sink, exists at a distance from other sources or sinks, the flow in the vicinity is in straight lines from or to the point, and these straight lines would be indefinitely extended if either source or sink existed by itself. As it is, the direction and amount of flow everywhere depends on the flow resulting from the whole arrangement of sources and sinks. Lines can be drawn in the medium which show the direction of the resultant flow from point to point, and these lines of flow can be so spaced as to indicate, by their closeness together or their distance apart, where the rate of flow is greater or smaller; and such lines start from sources, and either end in sinks or continue their course to infinity. In the electrical case these lines are the analogues of the lines of electric force (or field-intensity) in the insulating medium, which start from positive charges and end in negative, or are prolonged to infinity.

Across such lines of flow can be drawn a family of surfaces, to each of which the lines met by the surface are perpendicular. These surfaces are the equitemperature surfaces, or, as they are usually called, the isothermal surfaces. They can be drawn more closely crowded together, or more widely separated, so as to indicate where the rate of falling off of temperature (the "temperature slope") is greater or less, just as the contour lines in a map show the slopes on a hill-side.

Instead of the thermal analogy might have been used equally well that of steady flow in an indefinitely extended mass of homogeneous frictionless and incompressible fluid, into which fluid is being poured at a constant rate by sources and withdrawn by sinks. The isothermal surfaces are replaced by surfaces of equal pressure, while lines of flow in one are also lines of flow in the other.