the action of other molecules, and is therefore uniform and in a straight line. When however it comes very near another molecule, the two molecules act on each other for a very short time, the courses of both are changed and they go on in the new courses till they encounter other molecules. It would appear from the observed properties of gases that the mutual action between two molecules is insensible at all sensible distances. As the molecules approach, the action is at first attractive, but soon changes to a repulsive force of far greater magnitude, so that the general character of the encounter depends mainly on the repulsive force. On this theory, the elasticity of the gas may still be said in a certain sense to arise from the repulsive force between its molecules, only instead of this repulsive force being in constant action, it is called into play only during the encounters between two molecules. The intensity of the impulse is not the same for all encounters, but as it does not depend on the interval between the encounters, we may consider its mean value as constant. The average value of the force between two molecules is in this case the value of the impulse divided by the time between two encounters. Hence we may say that the force is inversely as the distance between the molecules, and that it acts between those molecules only which encounter each other. For an earlier investigation by Cavendish of the properties of an elastic fluid, see Note 18. NOTE 7, ART. 101. Here Cavendish endeavours to fix a precise meaning to the terms positively and negatively electrified," terms which he found current among electricians, but not well defined. The meaning which he here fixes to them, and which he afterwards makes much use of, is equivalent to the meaning of the modern term potential, as used by practical electricians. The idea of potential as used by mathematicians is expressed by Cavendish in his theory of canals of incompressible fluid. In the "Thoughts concerning Electricity," and in the unpublished papers, degrees of electrification are spoken of. These degrees of electrification are measured in the experimental researches by means of electrometers of different kinds, and since he has compared the indications of his electrometers with the degrees of electrification required to make a spark pass between the balls of Lane's discharging electrometer, we may express all these measurements in modern units, though Cavendish's original electrometers no longer exist. I have not been able to trace the idea of electric potential in the work of Epinus, so that Cavendish seems the first to have made use of it. The relation between the charge of a body and the degree of its electrification is the main object of Cavendish's experimental researches, and the results of his work were expressed in the material form of a collection of coated plates, each of which had a capacity equal to that of a sphere of known diameter. The leading idea in the great experimental work of Coulomb seems to be the measurement of the charges of the different bodies of a system and of parts of these bodies. Perhaps the most valuable of Coulomb's many contributions to experimental physics was the measurement of the surface-density of the distribution of electricity on a conductor on different parts of its surface by means of the proof plane. The numerical results obtained by Coulomb led directly to the great mathematical work of Poisson. I have not been able, however, to trace, even in those parts of Coulomb's papers where it would greatly facilitate the exposition, any idea of potential as a quantity which has the same value for all parts of a system of conductors communicating with each other. NOTE 8, p. 51. Cases of Attraction and Repulsion. The statements of Cavendish may be illustrated by the case of two spheres A and B, whose radii are a and b, and the distance between their centres c. If the charge of A is 1, and that of B is 0, the attraction is an expression which shows that it depends chiefly on the value of b, the radius of the sphere without charge. If the sphere B, instead of being without charge, is at potential zero, that is, if it is not insulated, the attraction is The number of times that the attraction of an uninsulated sphere exceeds that of a sphere without charge is therefore approximately c2 2 which is greater as the sphere is smaller. This agrees with what Cavendish says in Art. 108. With respect to two bodies at the same potential, Cavendish remarks in Art. 113, that it may be said that one of them may be rendered undercharged in the part nearest to the other, and he shows that even in this case, the two bodies must repel each other. But it may be shown that each of the bodies must be overcharged in every part of its surface. For in the first place no part can be undercharged, for the lines of force which terminate in an undercharged surface must have come from an overcharged surface at which the potential is higher than at the surface. But there is no body in the field at a higher potential than the two bodies considered. Hence no part of their surface can be undercharged. Nor can any finite part of the surface be free from charge, for it may be shown that if a finite portion of the surface of a conductor is free from charge, every point which can be reached by continuous motion from that part of the surface without passing through an electrified surface must be at the same potential. Hence no fiuite portion of a surface can be free from charge, unless the whole surface is free from charge. NOTE 9, ART. 124. The rate at which electricity passes from a conductor to the surrounding air or from the surrounding air to a conductor was believed to be much greater by Cavendish and his contemporaries than is consistent with modern experiments. Judging from the statements of the electricians of each generation, it would seem as if this rate had been diminishing steadily during the last hundred years in exact correspondence with the improvements which have been made in the construction of solid insulating supports for electrified conductors. Whenever the intensity of the electromotive force at the surface of a conductor is sufficiently great, the air no doubt becomes charged.* This is the case at a sharp point connected with the conductor even when the potential is low, but when the curvature of the surface is continuous and gentle, the conductor must be raised to a high potential before any discharge to air begins to take place. Thus in Thomson's portable electrometer, in which there are two disks placed parallel to each other at different potentials, the percentage loss of electricity from day to day is very small, and seems to depend principally on the solid insulators, for when the disks are placed very near each other, less loss is observed than when they are further apart, though the intensity of the force urging the electricity through the intervening stratum of air is greater the nearer the disks are to each other. * M. R. Nahrwold (Wiedemann's Annalen v. (1878) p. 440) finds that the discharge from a sharp point communicates a charge to dusty air which can be detected in the air for some time afterwards. This does not occur in air free from dust. But the discharge from an incandescent platinum wire communicates a lasting charge even to air free from dust. On the surface density of electricity near the vertex of a cone. Green has given in a note to his Essay, section (12), the following results of an investigation which, so far as I am aware, he never published. "Since this was written, I have obtained formulæ serving to express, generally, the law of the distribution of the electric fluid near the apex 0 of a cone, which forms part of a conducting surface of revolution having the same axis. From these formulæ it results that, when the apex of the cone is directed inwards, the density of the fluid at any point p, near to it, is proportional to -1; r being the distance Op, and the exponent n very nearly such as would satisfy the simple equation (4n+2) B = 3π, where 2ẞ is the angle at the summit of the cone. If 2ẞ exceeds π, this summit is directed outwards, and when the excess is not very considerable, n will be given as above: but 2ẞ still increasing, until it becomes 2 - 2y, the angle 2y at the summit of the cone, which is now directed outwards, being very small, n will be given by and in case the conducting body is a sphere whose radius is b, on which P represents the mean density of the electric fluid; p, the value of the density near the apex O, will be determined by the formula Professor F. G. Mehler* of Elbing has investigated the distribution of electricity on a cone under the influence of a charged point on the axis, and the inverse problem of the distribution on a spindle formed by the revolution of the segment of a circle about its chord. He finds that when the segment is a very small portion of the circle, so that the conical points of the spindle are very acute, the surfacedensity at any point is inversely proportional to the product of the distances of that point from the two conical points. Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function, und ihre Anwendung in der Theorie der Electricitätsvertheilung. (Elbing, 1870.) NOTE 10, p. 63. Sir W. Thomson* has determined in absolute measure the electromotive force required to produce a spark in air between two electrodes in the form of disks, one of which was plane, and the other slightly convex, placed at different distances from each other. Mr Macfarlane + has recently made a more extensive series of experiments on the disruptive discharge of electricity. He finds that in air at the ordinary pressure and temperature the electromotive force required to produce a spark between disks, 10 cm. diameter, and from 1 to 0.025 cm. apart, is expressed by the empirical equation V = 66·940 (s2 + -205038)*, where 8 is the distance between the disks. If we suppose that in the space between the disks the potential varies uniformly, as it does between two infinite planes, then the resultant electromotive intensity is R= 8 V If, on the other hand, we suppose that the variation of the potential near the surface of the disks is affected by unknown causes, we would get a better estimate of the intensity by taking V Both dV and diminish as the distance increases, approximating to the limit 66.940. 8 This corresponds to a surface-density of 5-327 units of electricity per square centimetre, and to a tension of 178.3 dynes per square centimetre. As the ordinary pressure of the atmosphere is about a million dynes per square centimetre, the pressure with which the electricity tends to break through the air is only about pressure of the atmosphere. 1 5600 of the If the electrodes are convex surfaces, whose radii of curvature, a and b, are large compared with the least distance c between the surfaces, then if the greatest electric force at the surface whose radius is a will be equal to that at either of two parallel plane surfaces at the same potentials whose distance is s. * Proc. R. S., 1860, or Papers on Electrostatics, chap. XIX. |