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.