OOULOIIB S RESULTS.] E L E G T 11 I C I T T 23 be made large, and the surface density on its outside there fore small, there will be little loss of the external charge ; and the method has this advantage, that dissipation from the proof-plane inside the cage does not atfect the result of the measurement in hand, it being indifferent, qua effect on the electrometer, whether the electricity inside the cage be on the proof- plane, in the air, or elsewhere, provided merely it be inside. The state of the cage as to electrified air, &c., is easily tested by the electrometer at any time. Coulomb s Results. If we electrify a sphere, and test the electrical density at two points of its surface, experiment will show, as would be expected from the symmetry of the body, that the density at the two points is the same. If we test the electric density at any point of a sphere, and then halve its charge by division with an equal neutral sphere, and test the electric density again, we shall find it half what it was before. The electric density at any point is therefore propor tional to the whole charge on the sphere, or to the mean density, meaning by that the whole charge divided by the whole surface of the sphere. If, instead of a sphere, we operate with an ellipsoid generated by the revolution of an ellipse about its major axis, we shall lind that the electric density is not uniform as in the case of the sphere, but greater at the sharp ends of the major axis than at the equator, and the ratio of the densities increases indefinitely as we make the ellipsoid sharper and sharper. This leads us to state a principle of great importance in the theory of electrical distribution, viz., that the electrical density is very great at any pointed part of a conductor. If we determine the ratio of the densities at two points of an ellipsoid, 1 diminish the charge, and redetermine the same ratio, we shall find that, although the actual densities are diminished, the ratio remains the same ; and if we determine the density at any point of the ellipsoid, and then halve its charge b^ touching it with an equal and similar ellipsoid (they must be placed with their axes in the same straight line, and made to touch at the poles), 2 and redetermine the density at the same point as before, we shall find that the density in the second case is half that in the first. We have in fact, in general, the im portant proposition that The density at any point of a conductor is proportional to the whole charge on the conductor, or, ivhat is the same, to the mean density. The following case given by Coulomb is interesting ; it shows the tendency of electricity towards the projecting parts, ends, or points of bodies. The conductor was a cylinder with hemispherical ends, the length of the cylin der being 30 inches, its diameter 2 inches. Coulomb gives the following results : Distance from end. Density. 5 in. 1-00 2 1-25 1 1-80
2-30 The density at the end is thus more than twice that at the middle. Other results, taken from Coulomb s unpublished papers, may be found in Biot, 3 Mascart, or Riess. His results for a circular disc we shall quote further on. 1 We suppose in all these experiments that we are dealing with a single body, sufficiently distant not only from all electrified bodies but from all neutral conductors to be undisturbed by them. This condi tion is essential. 2 It would not do to make the pole of one touch the equator of the other, or to place them otherwise unsyrmnetrically 8 Traitt de Physique. Riess made a series of experiments on cubes, cones, <fcc. j but as these are not of theoretical interest, the calculation in such cases being beyond the powers of analysis at present, and as the use of the proof-plane or sphere with bodies where edges and points occur is not free from objection, we content ourselves with referring to Riess s work for an account of the results. Coulomb made a series of experiments on bodies of Ccm- different forms, which he built up out of spheres of different lomb s sizes, or out of spheres and cylinders. These are of very re great interest, partly on account of the close agreement of on com _ some of the results with the deductions subsequently made posite by Poisson from the mathematical theory, and partly on conduo- account of the clearness with which they convey to the tors> mind the general principles of electric distribution. His method in most cases was to build up the conductor and electrify it with all the different parts in contact, and then after separating the parts widely, to determine the mean density or the whole amount of electricity on each part by the proof-plane or otherwise. For spheres in contact he found the following results, S, Q, cr; S , Q , cr denoting the surface, quantity of elec tricity, and mean surface density for the two spheres respec tively. S 3 a s Q cr 3-36 3-8 1-09 14-80 11-1 1-33 62-00 37-6 1-65 From this it appears that although the whole amount of electricity on the large sphere is greater than that on the small, yet the mean density for the smaller sphere is greater than for the larger. The above result also affords an experimental illustration of the action of the earth in discharging a conductor connected with it. Comparing the conductor to the small sphere and the earth to the large sphere of 62 times the superficial area of the small one, if we start with charge Q on small sphere and then put the two in contact, the charge on the small sphere will be reduced to Q, so that the mean density is dimin- OO D ished in the ratio 1 : 38 - 6. This ratio increases indefinitely S as the ratio increases. These results are in satisfactory o agreement with Poisson s calculations. Coulomb was led by his observations to assign 2 as the limit of the ratio of the mean densities when the ratio of the diameters of the spheres is infinitely great ; the mathematical theory gives ^ or 1-65. u Coulomb also determined the density at the apex or smaller end of the body formed by two unequal spheres in contact. The following are his results, the mean density of the larger sphere being unity : Density at apex. Observed. Calculated. 1 1-27 1-32 2 1 55 1-83 4 2-35 2 48 8 3-18 3-09 8 4-00 4-21 When two equal spheres are placed in contact the dis tribution will of course be the same in each; Coulomb found that, from the point of contact up to a point on the surface of either sphere distant from it by about 20, no
trace of electricity could be observed ; at 30, GO", 90,