equilibrium at other distances. We thus obtain variable distances between the balls, and angles of torsion corresponding to these distances; and it is from the relations that exist, on the one hand between these distances, and on the other between these angles, that we deduce, as in the case of repulsion, the law that we have laid down. We must only take care that, in making the torsions vary, we do not bring the two balls at so feeble a distance from each other that, attraction getting the better of the force of torsion, they may come suddenly into contact, in which case, the two electricities being neutralised, all would have to be done over again.* In all the preceding experiments the electricity that is possessed by the balls runs the risk of being dissipated, with greater or less rapidity, by means of the imperfect insulation of the supports and the humidity of the air. To avoid this inconvenience, we must dry the interior of the torsion balance as much as possible, by placing in it chloride of calcium or other bodies that absorb moisture, by making the experiment as rapidly as possible, and finally by taking account of the loss of electricity, by taking the mean between the results of the same experiments made at slightly different periods.† Influence of the absolute Quantity of Electricity upon Attractions and Repulsions; and general Expression of the Attractive and Repulsive Force. After having found the laws by which electric attractions and repulsions are connected with the mutual distance of two electrised bodies, it remained to determine the law according to which the attractive or repulsive force depends on the *For the calculations, see note A at the end of the volume. † Coulomb confirmed the laws that he had found by the torsion balance by a totally different method, which consists in oscillating before an insulated and electrised globe, a horizontal needle terminated by a small ball, charged with the same electricity as the globe, or with a different electricity; the number of oscillations in a given time at different distances are counted; and, by the formula of the pendulum, is deduced the influence of distance upon the intensity of the force. We shall develope this method more in detail when we come to magnetic attractions and repulsions. quantities of electricity accumulated upon each body. To arrive at this, Coulomb set out from the self-evident principle that if two bodies, for example, two insulated conducting spheres, of the same size and perfectly similar in every respect, are placed in contact, they share equally in the electricities that they possess; in such sort that if one of the insulated spheres is electrised and the other is not, they have, after contact, each the same quantity of electricity, namely the half of that possessed beforehand by the one only that was electrised. This point being admitted, we observe the force of torsion that at a certain distance is in equilibrium with the repulsive or attractive force of the two balls of the balance, that are similar and charged with the same quantity of electricity. We have a third ball perfectly similar to the two others, insulated as they are, but not electrised. With this ball we touch the 'fixed ball of the balance; this contact takes from it the half of its electricity according to the principle we have just laid down, the movable ball retaining the whole of its own. We then look again for the force of torsion necessary to cause equilibrium, at the same distance, to the attractive or repulsive force of the two balls of the balance, and we find that this force is now only the half of what it was before. By then reducing to a half, by the same process, the electricity of the movable ball, we find that the force of torsion is now only the fourth of what it was in the outset. It is the same if, without making any change in the electric state of the movable ball, we diminish a second time by a half the electricity of the fixed ball, namely, if we reduce it to a fourth of what it was at the first. These experiments therefore prove that, the distance remaining the same, the attractions and repulsions are in compound ratio to the quantities of electricity with which the two bodies are charged; or, which amounts to the same thing, that the attractive or repulsive force is the product of these two quantities. It is easy indeed to see that there is simply a product which may become half less when one of the factors diminishes by at half, and become four times less when the two factors each diminish by a half, or when one alone of the two becomes a fourth of what it was, the other not changing. This result has been verified by Coulomb, by means of a great number of experiments, made with absolute quantities of electricity, and very different one from the other. EE' The law that we have just established, connected with that which regulates the distance, enables us to give to the expression of the attractive and repulsive force reigning between two electrised bodies this very simple form: F= D2, calling F the force, E and E' the quantities of electricity with which the two bodies are charged, and D the distance existing between them. Torsion Electrometers. The knowledge of the two laws to which electric attractions and repulsions are subjected, has furnished philosophers with an excellent electrometer in the torsion balance. To apply it to this use the fixed ball is made to communicate, by means of a metal rod situate in the axis of the insulating tube that sustains it, with a small metal sphere situated on the outside of the glass cage that contains the whole apparatus. This sphere, as in ordinary electroscopes, is placed in communication with the source of electricity. The electricity arrives at the fixed ball and at the movable one that is in contact with it at the 0° of torsion and charges each of them. Repulsion immediately takes place; by means of torsion they are to be brought back to a determinate distance, always the same for comparative experiments. The angles of torsion, necessary to bring back the balls to the constant distance, represent in each experiment the repulsive forces that are proportional to these angles. But these forces being the product of the two equal quantities of electricity with which the balls are charged, it is clear that each of these quantities themselves is proportional to the square roots of the repulsive forces or of the corresponding angles of torsion. We have, in fact, in this E2 case, F = and in the other F': E/2 = whence E E':: F D29 F'. Thus, if the angle is four times greater in one experiment than in the ather, it signifies that the quantity of electricity possessed by each of these balls, and consequently that of the source which was put in communication with them, is twice greater. Care must be taken after each experiment, and before commencing another, to discharge the two balls of the balance, by putting them in communication with the ground by means of a metal rod held in the hand. Without this precaution, the electricity that would remain after one experiment would complicate and falsify the results of the following one. It is true we need not discharge the movable ball, but leave to it a constant quantity of electricity, taking care to avoid its coming in contact with the fixed ball. Then this latter alone is placed in communication with the source; and the angles of torsion or the repulsive forces are then simply proportional to the quantities of electricity with which it is charged, quantities which are variable in this ball alone. This mode of operating is easier, and it requires no calculation; hence it is more adopted, although it is less sure on account of the loss of electricity which the movable ball always experiences, to a greater or less extent, during the experiments, notwithstanding our carefully taking the precautions that we have already pointed out when referring to the torsion balance. Coulomb's electric balance, when employed as an electrometer, does not, it is true, immediately indicate the nature of the electricity; a determination, however, which may easily be obtained by the same means employed for other electroscopes. But the great superiority of this instrument is that it is the only one which can give an exact measure of electric forces; in addition to which, it is susceptible of great sensibility. When necessary to increase this sensibility, the metal wire, by the torsion of which the forces are to be measured, must be as fine as possible, and as long as the construction of the apparatus permits. Care is taken at the same time that the glass or gum-lac needle that carries the movable ball as well as this ball itself, are of great lightness. By these means we succeed in obtaining an apparatus of remarkable delicacy. It may even become the most delicate of all electroscopes, except, perhaps, that with dry piles, if a film unwound from a cocoon is put in place of the metal wire that carries the movable needle: we then give it a slightly different and more simple form (Fig. 38.). But as the waxed filament of silk does not obey the laws of torsion as the metal wires do, the apparatus is no longer an electrometer; it becomes a simple electroscope. In this case a force equal to th of a grain is sufficient to make the needle traverse Fig. 38. an entire circumference; consequently, an arc of one degree, if traversed by this needle, corresponds to a force equivalent to only 21600000 of a grain. We see, by this example, what a minute force we can contrive to measure, and, consequently, to what extent we may succeed in discovering the slightest traces of electricity. Objections to the Generality of the preceding Laws. Before terminating this chapter, we must add that the generality of the two laws discovered by Coulomb has been contested by an English philosopher, Sir Wm. Snow Harris. This philosopher has made a great number of observations by means of an apparatus of his invention, termed a Bifilar balance, in which the movable needle is carried by two waxed silk threads, the points of support of which are very near to each other, and at an equal distance from its centre of gravity. As soon as the movable needle is driven from its position of equilibrium, the two threads can no longer preserve their vertical position, and they incline in opposite directions to a greater or less degree, according to the intensity of the force by which the needle is driven; and it necessarily follows that the latter is raised. The new position that it acquires is, therefore, that at which there is equilibrium between the electric force and the force with which gravity tends to bring it back to its normal position, a force which it is easy to calculate. In Sir W. Harris's bifilar balance, gravity takes the place of the force of torsion in Coulomb's balance. With regard to the slight torsion that the silk threads may undergo, |