It was there pointed out that an electrometer is essentially an instrument for measuring differences of electric potential between conductors, by means of effects of electrostatic force. Such a difference is what a gold-leaf electroscope indicates for its gold leaves and the walls surrounding the air-space in which they are suspended. As electroscopes used to be constructed, these walls were made of glass imperfectly covered, if at all, by conducting material, and the electroscope was quite indefinite and uncertain in its action. The instrument was also, as made, quite insensitive. Recently, however, it has been rehabilitated in reputation, and brought into use as a very sensitive indicator of effects of radio-activity.

Thomson described in this paper six species of electrometers of his own devising. The best known of these are his quadrant electrometer and his attracted-disk electrometers. The former is to be found in some form or other in every laboratory nowadays, and need not be described in detail. The action is of two conductors—the two pairs of opposite quadrants of a shallow, horizontal, cylindrical box, made by dividing the box into four by two slits at right angles—upon an electrified slip of aluminium suspended by a two-thread suspension within the box, with its length along one of the slits. The two pairs of opposite quadrants are at the potential difference to be measured, and the slip of aluminium, or "needle," has each end urged round from a quadrant at higher potential towards one at a lower, and these actions conspire to turn the slip against its tendency to return to the position in which the two threads are in one plane. Thus the deflection (measured by the displacement of a reflected ray of light used as index) gives an indication of the amount of the potential difference.

The electrification of the "needle" was kept up by enclosing the quadrantal box within an electrified Leyden jar, to the interior coating of which contact is made by a platinum wire, depending from the needle to sulphuric acid contained in the jar. The whole apparatus was enclosed in a conducting case connected to earth. This made its action perfectly definite. Variations of this electrification of the jar were shown by an attached attracted-disk electrometer, the principle of which we shall merely indicate.

The quadrant electrometer has now been vastly increased in sensibility by the use of a single quartz fibre as suspension. By the invention of this fibre, which is exceedingly strong and is, moreover, so definite in its elastic properties that it comes back at once exactly to its former zero state after twist, Mr. C. V. Boys has increased the delicacy of all kinds of suspended indicators many fold. But it ought to be remembered that a Dolezalek electrometer, with some hundred or more times the sensibility of the bifilar instrument, was only made possible by its predecessor.

Attracted-disk-electrometers simply measure, either by weighing or by the deflection of a spring, the attractive force between two parallel disks at different potentials. From the determination of this force, and the measurement of the distance between the disks (or better, of an alteration of the distance) a difference of potentials can be determined, and a unit for it obtained, which is in direct and known relation to ordinary dynamical units. Thomson's "Absolute Electrometer" was designed specially for accurate determinations of this kind. Another form, called the Long Range Electrometer, was devised for the measurement of the potentials of the charged conductors in electric machines and Leyden jars.

Accurate determinations of the sparking resistance between parallel plates charged to different potentials in air were made by means of attracted-disk-electrometers in the course of some important experiments described in the Electrostatics and Magnetism. These results have been much referred to in later researches.

A small attracted-disk-electrometer was used as indicated above to keep a watch on the electrification of the Leyden jar of the quadrant instrument, and a small induction machine was added, by turning which the operator could make good any loss of charge of the jar.

This electrical machine was an example of an apparatus on precisely the same principle as the Voss or Wimshurst machines of the present day. In it by a set of moving carriers, influenced by conductors, the charges of the latter were increased according to a compound interest principle only interfered with by leakage to the air or by the supports. Several forms of this machine, on the same principle, were constructed by Thomson, and described in 1868; but he afterwards found that he had been anticipated by C. F. Varley in 1860. Still later it was discovered that a similar instrument had been made a century before by Nicholson, and called by him the "Revolving Doubler."

The experiments which Thomson made on atmospheric electricity at the old College tower, and by means of portable electrometers in Arran and elsewhere, can only be mentioned. They led no doubt to some improvements on electrometers which he made, the method of bringing the nozzle of a water-dropper, or a point on a portable electrometer to the potential of the air, by the inductive action on a stream of water-drops in the one case, or the particles of smoke from a burning match in the other. He invented a self-acting machine, worked by a stream of water-drops, for accumulating electric charges, on the principle of the revolving doubler. It was this apparently that led to the machines with revolving carriers, to which reference has been made above.

The mathematical theory of magnetism which Thomson gave in 1849, in the Phil. Trans. R.S., was, when completed by various later papers, a systematic discussion of the whole subject, including electromagnetism and diamagnetism. To a large extent the ground covered by the 1849 paper had been traversed before by Poisson, and partially by Murphy and Green; but Thomson stated that one chief object of his memoir was to formally construct the theory without reference to the two magnetic fluids, by means of which the facts of experiment and conclusions of theory had so far been expressed. He found it, however, convenient to introduce the idea of positive and negative magnetic matter (attracting and repelling as do charges of positive and negative electricity), which are to be regarded as always present in equal amounts, not only in a magnet as a whole, but in every portion of a magnet; and at first sight this might appear like a return to the magnetic fluids. But it amounts on the whole rather to a conception of a magnet as a conglomeration of doublets of magnetic matter (that is, very close, equal and inseparable charges of the two kinds of matter), the arrangement of which can be changed by the action of magnetic force. This idea is set forth now in all the books on magnetism and electricity. There can be no doubt that the systematic presentment of the subject by Thomson, and the theorems and ideas of magnetic force and magnetic permeability by which he rendered the clear, and therefore mathematical, notions of Faraday explicitly quantitative, had much influence in furthering the progress of electrical science, and so leading on the one hand to the electromagnetic theories of Maxwell, and on the other to modern research on the magnetic properties of iron, and to the correct ideas which now prevail as to construction of dynamo-electric machines and motors.