Determining A, so as to make Л(^,+λ) (4+41⁄2)dæ a minimum, we find 2 0 This approximation is evidently of little use unless the length of the cylinder considerably exceeds 7.245 times its diameter, for this ratio makes the second term of the denominator infinite. It shows, however, that when the ratio of the length to the diameter is very great, the true capacity approximates to the value of K ̧ given in (18). We may proceed in the same way to determine A, and 4, so that 2 shall be a minimum, and we thus find a third approximation to the value of the capacity, in which so that when £ is very large the distribution approximates to The value of the inferior limit of the capacity, as given by this approximation is As £ increases, K approaches to the value found by the first approxi mation. To indicate the degree of approximation, the value of £ and of the successive terms of the denominator are given below. The observed capacities of Cavendish's cylinders may be deduced from the numbers given in Art. 281 by taking the capacity of the globe of 12.1 inches diameter equal to 6.05, and their capacities as calculated by the formula of this note are given in the following table. The agreement of the calculated and measured values is remarkable. NOTE 13, ARTS. 152, 280. Two cylinders. In the case of two equal and parallel cylinders at distance c, the linear densities being uniform and equal to λ, and A, the part of the potential energy arising from their mutual action is If the two cylinders are in electric communication with each other A,A,, and the capacity of the two cylinders together is approximately If a cylinder is placed at a distance d from a conducting plane surface and parallel to it, then the electric image of the cylinder will be at a distance c = 2d, and its charge will be negative, so that the capacity 401 of the cylinder will be increased. The capacity of the cylinder in presence of a conducting plane at distance c, is log 1 - log +27 r-c + с 21 41 b Thus in Cavendish's experiment he used a brass wire 72 inches long and 0.185 in diameter. tance from any other body would be 5.668 inches. Cavendish placed it The capacity of this wire at a great dishorizontally 50 inches from the floor. would increase its capacity to 5.994 inches; Cavendish, by comparison The inductive action of the floor with his globe, makes it 5-844. To compare with this he had two wires each 36 inches long and 0.1 inch diameter. The capacity of one of these at a distance from any other body would be 2-8697 inches, or the two together would be 5.7394 inches. The two wires were placed parallel and horizontal at 50 inches from the floor. Each wire was therefore influenced by the other wire, and also by the negative images of itself and the other wire. The denominator of the fraction expressing the capacity is therefore Distance. 18 24 36 Wire Other itself. wire. Own image. Other image. 6.27249.8256 0.1759 - 0·1754 6.2724+0.6596 – 0·1759 0·1733 6.5828 The numerator of the fraction which expresses the capacity of both wires together is 36, so that the capacity of the two is If we suppose the plate AB to be overcharged and the plate DF to be equally undercharged, the redundant fluid in any element of AB being numerically equal to the deficient fluid in the corresponding element of DF, then what Cavendish calls the repulsion on the column CE in opposite directions becomes in modern language the excess of the potential at C over that at E. determine approximately the difference of the potentials of two curved Hence the object of the Lemma is to plates when their equal and opposite charges are given, and to deduce their charges when the difference of their potentials is given. M. NOTE 15, Art. 169. On the Theory of Dielectrics. Cavendish explains the fact discovered by him, that the charge of a coated glass plate is much greater than that of a plate of air of the same dimensions, by supposing that in certain portions of the glass the electric fluid is free to move, while in the rest of the glass it is fixed. Probably for the sake of being able to apply his mathematical theorems, he takes the case in which the conducting parts of the glass are in the form of strata parallel to the surfaces of the glass. He is perfectly aware that this is not a true physical theory, for if such conducting strata existed in a plate of glass, they would make it a good conductor for an electric current parallel to its surfaces. As this is not the case, Cavendish is obliged to stipulate, as in this proposition, that the conducting strata conduct freely perpendicularly to their surfaces, but do not conduct in directions parallel to their surfaces. The idea of some peculiar structure in plates of glass was not peculiar to Cavendish. Franklin had shewn that the surface of glass plates could be charged with a large quantity of electricity, and therefore supposed that the electric fluid was able to penetrate to a certain depth into the glass, though it was not able to get through to the other side, or to effect a junction with the negative charge on the other side of the plate. The most obvious explanation of this was by supposing that there was a stratum of a certain thickness on each side of the plate into which electricity can penetrate, but that in the middle of the plate there was a stratum impervious to electricity. Franklin endeavoured to test this hypothesis by grinding away five-sixths of the thickness of the glass from the side of one of his vials, but he found that the remaining sixth was just as impervious to electricity as the rest of the glass*. It was probably for reasons of this kind, as well as to ensure that his thin plates were of the same material as his thick ones, that Cavendish prepared his thin plate of crown glass by grinding equal portions off both sides of a thicker plate. [Art. 378.] It appears, however, from the experiments, that the proportion of the thickness of the conducting to the non-conducting strata is the same for the thin plates as the thick ones, so that the operation of grinding must have removed non-conducting portions as well as conducting ones, and we cannot suppose the plate to consist of one non-conducting stratum with a conducting stratum on each side, but must suppose that the conducting portions of the glass are very small, but so numerous that they form a considerable part of the whole * Franklin's Works, 2nd Edition, Vol. 1. p. 301, Letter to Dr Lining, March 18, 1755. volume of the glass. If we suppose the conducting portions to be of small dimensions in every direction, and to be completely separated from each other by non-conducting matter, we can explain the phenomena without introducing the possibility of conduction through finite portions of glass. It was probably because Cavendish had made out the mathematical theory of stratified condensers, but did not see his way to a complete mathematical theory of insulating media, in which small conducting portions are disseminated, that he here expounds the theory of strata which conduct electricity perpendicularly to their surfaces but not parallel to them. In forming a theory of the magnetization of iron, Poisson was led to the hypothesis that the magnetic fluids are free to move within certain small portions of the iron, which he calls magnetic molecules, but that they cannot pass from one molecule to another, and he calculates the result on the supposition that these molecules are spherical, and that their distances from each other are large compared with their radii. When Faraday had afterwards rediscovered the properties of dielectrics, Mossotti, noticing the analogy between these properties and those of magnetic substances, constructed a mathematical theory of dielectrics, by taking Poisson's memoir and substituting electrical terms for magnetic, and Italian for French, throughout. A theory of this kind is capable of accounting for the specific inductive capacity being greater than unity, without introducing conductivity through portions of the substance of sensible size. Another phenomenon which we have to account for is that of the residual charge of condensers, and what Faraday called electric absorption. The only notice which Cavendish has left us of a phenomenon of this kind is that recorded in Arts. 522, 523, in which it appeared "that a Florence flask contained more electricity when it continued charged a good while than when charged and discharged immediately." To illustrate this phenomenon, I gave in "Electricity and Magnetism," Art. 328, a theory of a dielectric composed of strata of different dielectric and conducting properties. Professor Rowland has since shown* that phenomena of the same kind would be observed if the medium consisted of small portions of different kinds well mingled together, though the individual portions may be too small to be observed separately. It follows from the property of electric absorption that in experiments to determine the specific inductive capacity of a substance, the result depends on the time during which the substance is electrified. Hence most of those who have attempted to determine the value of this quantity for glass have obtained results so inconsistent with * American Journal of Mathematics, No. I. 1878, p. 53. |