Thus the increase with the temperature is very considerable. The linear coefficient of expansion for the temperature t can be put 0.000061+0-00000076t. Two strips of ebonite and sheet zinc, when rivetted together, curve very materially with even a moderate heating. A thin strip of ivory 20 centims. in length, cemented by means of isinglass to a similar one of ebonite, even without index, forms a delicate thermometer, seeing that its free end moves through several millimetres for one degree. The curvature in consequence of unequal expansion may be most simply demonstrated by the aid of a mere plate of ebonite for, owing to its bad conductivity, it curves considerably when rapidly heated on one side. The solid expansion of ebonite is, from the above numbers, at 0° equal to that of mercury; at higher temperatures it is still greater. It is possible that other kinds expand still more, so that as a curiosity a mercurial thermometer might be constructed whose scale sunk on being heated. 串 ; The great expansion may possibly be connected with the proportion of sulphur which ebonite contains: Kopp found for the coefficient of sulphur 0.000061 at 30°. On the other hand, the contrast to soft caoutchouc is very remarkable. I will mention one fact which was observed in the observation of expansion. The bar of ebonite, which was about a centimetre in thickness, after being heated required a considerable time before it assumed a constant length. Although the bad conductivity is doubtless the principal cause of this, I imagine that another phenomenon is also at work. Like the elastic change of form, the expansion by heat may also not take place instantaneously, but continue itself after the change of temperature, gradually becoming weaker. A few observations of Matthiessen's with glass rods (Poggendorff's Annalen, vol. cxxviii. p. 521) seem to point in this direction; probably this thermal afteraction, like the elastic, occurs in an eminent degree in organic substances.-Poggendorff's Aunalen, No. 8, 1873. ON THE DISCHARGE OF ELECTRIFIED CONDUCTORS. BY J. MOUTIER. : Equilibrium of electricity at the surface of a system of conductors results, as Poisson has shown, from the following condition :The resultant of the actions exerted by the different electric layers on any point in the interior of the conductors must be no action. If by m be designated one of the electric masses, affected by a sign according to its nature, by r its distance from a point situated within the conductor, the function * Pogg. Ann. vol. lxxxvi. p. 156. must then have a constant value for all points situated within one and the same conductor. The function V, introduced into analysis by Laplace, has been designated by Green under the name of the potential function. If m and m' denote two masses of electricity of the same name situated at a distance r, each of the masses is repelled by a force mm' equal to ; if the distance of the masses becomes r+dr, the sum of the elementary works of the repulsive forces is mm' dr-d ·a(mm'). The sum of the elementary works of all the repulsive forces is therefore equal to the increase of the function in which the summation is extended to all the masses of electricity, each of them being supposed to be affected by a sign. The function W is designated, according to a notation borrowed from Gauss, by the name of the potential of electricity. The works of Helmholtz and Clausius have particularly called attention to this function, which plays a considerable part in the phenomenon of the electric discharge; indeed the increase of the potential represents the work done in the electric discharge. M. Clausius has shown that the potential can be readily expressed by means of the charges and potential functions relative to each of the conductors. If V is the potential function on one conductor, and Q its charge, W=VQ. If Vo, Q, are the initial values, V1, Q, the final values relative to one conductor, W1-W=VQ。— £ &V,Q, measures the work done in the partial discharge of the system of conductors; and when the conductors are restored to the neutral state, the work of the complete discharge is ZV.Q.• The mechanical equivalent of the discharge is independent of the manner in which the discharge is effected; it depends only on the initial and final values of the potential; so that the sum of the effects of the electrical discharge remains the same, whatever may be the nature of the discharge. M. Helmholtz has already estimated the vis viva gained by the electricity in passing from the surface of a conductor to an infinite distance; but he considered the potential of the electricity of a conductor a constant quantity, while in reality the potential on the conductor diminishes in proportion to the charge: this diminution has the effect of modifying the expression of the work produced in the discharge through air. dV Let us consider an electrified conductor having a charge q, and suppose that a quantity of electricity dq escapes from the electrified body and disappears into the air. When this electricity dq passes from a level surface where the potential function has a value V, to the infinitely near level surface where the potential function has the value V+du, designating by dn the infinitely small portion of normal comprised between the two surfaces at the point considered, the repulsive force exerted on dq is · dq. The elementary work of the repulsion, in passing from one level surface to the surface infinitely near is -dVdq. Consequently, when the quantity of electricity dq removes to infinity, the corresponding work has the value V'dq, if we call V' the potential function at the surface of the conductor, or, what is the same thing, in the interior of this conductor: this expression has already been given by M. Helmholtz. dn But in proportion as the loss of electricity is effected, the charge on the conductor diminishes; it is the same with the potential function. The potential function V' is proportional to the charge q of the conductor; we can put V'=aq, a being a constant peculiar to the conductor. The work necessary to repel to infinity the quantity of electricity dq is aq dq. Consequently, calling the initial charge of the conductor q, the work necessary for repelling all the electricity of the body to an infinite distance has for its expression V, denoting the initial value of the potential function on the conductor. Therefore the work consumed by the repulsion of all the electricity to infinity is equal to the potential of the electricity; or, in other terms, the mechanical equivalent of the external discharge is equal to the potential of the electricity. The result would evidently be the same for a system of electrified conductors. The value of T is independent of the path pursued by the electricity which escapes from the conductors; it is easy to recognize that this value remains the same when two equal quantities of opposite electricities meet on their passage and recombine. Let us in fact suppose that two quantities of electricity, +m and -m, recombine in a point M to form neutral electricity; let V be the potential function in this point. Suppose that at the point M dV the electricity +m is repelled by the force-m; the portion of the work T necessary to remove +m to infinity, starting from the point M, is Vm. The quantity -m, on the contrary, situated at the point M, is attracted; the attractive force has the same value as the repulsive force; and when -m removes from M to infinity, the corresponding work is equal to the preceding and of the contrary sign. The value of T remains consequently the same, whether the two electricities recede to infinity or recombination takes place at any point whatever. It is necessarily the same if recombination is produced upon a conductor. This is the case presented, for instance, in the discharge from a Leyden jar. Let us consider a spherical jar: we will call r the radius of the sphere which is formed by the interior coating, e the thickness of the glass. If q denotes the charge of the interior coating at a given instant, dq the quantity of electricity repelled from the interior to the exterior coating when these two coatings are united by a conductor, the work effected in the repulsion of the quantity of electricity dq is The work of the repulsion which corresponds to the quantity of electricity q, originally contained on the interior coating is The factor contained in the parenthesis represents the potential function on the interior coating. We thus again find, in the particular case of the Leyden jar, the expression of the potential of the electricity. M. Helmholtz was the first who applied the theory of the potential to the discharge of the Leyden jar. His researches have been completed by M. Clausius; and the theory of the experiments of M. Riess can now be regarded as very satisfactory. It still remains, however, to inquire how the discharge is produced, independently of the value of its mechanical equivalent. M. Helmholtz, after explaining the heat disengaged in M. Riess's experiments, adds: "This law is easy to understand, provided the discharge of a battery be not represented as a simple movement of electricity in one direction, but AS A SERIES OF OSCILLATIONS BETWEEN THE TWO COATINGS, oscillations which become less and less continually until the vis viva is extinguished by the sum of the resistances". We have just seen that the discharge may be represented by a movement of the electricity directed from one coating towards the other. : From the preceding may be deduced the demonstration of a theorem established by Gauss in the case of a single conductor, and generalized afterwards by M. Liouville for a system of conductors :When conductors contain respectively equal quantities of the two fluids, all these conductors are in the neutral state. Indeed in this case the potential is nil; consequently the external discharge of the system of conductors cannot give rise to any work.-Comptes Rendus de l'Académie des Sciences, Nov. 24, 1873, pp. 1238–1241. 45. *Théorie Mécanique de la Chaleur, traduite par M. F. Folie, vol. ii. p. † Mémoire sur la conservation de la force, traduit par L. Pérard, p. 107. THE LONDON, EDINBURGH, AND DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [FOURTH SERIES.] MARCH 1874. XXI. On the Electric Resistance of Selenium. THE recently discovered fact of the diminution of the electric resistance of selenium in the crystalline state when exposed to the action of light or, possibly, of radiant heat, is one which naturally excites some interest beyond that arising from the curious and unexpected nature of the phenomenon considered by itself; for the possibility of selenium being applied to the measurement of light or radiant heat invests the discovery with a very general importance. Mr. Willoughby Smith seems to have satisfied himself that light, not heat, is the active agent; but I have spoken of the latter as possibly the cause of the effect observed, as Lieutenant Sale's paper in the Proceedings of the Royal Society (although it is therein stated that selenium is affected by light, and again, that the change of resistance is not due to an alteration of temperature) might lead one to infer that the observed effect was due to radiant heat, not to light; for he says that the actinic rays produce no effect, but that it is at a maximum in the red rays, or beyond them, near the maximum of the heat-rays; and inasmuch as he appears not to have determined by means of the thermopile the relative calorific power of the various rays of his spectrum, nor even to have reduced his results to what they would have been if the normal or diffraction spectrum had been employed, the experiment is inconclusive as to the comparative * Communicated by the Author. Phil. Mag. S. 4, Vol. 47. No. 311. March 1874. M |