Page:Reprint of Papers on Electrostatics and.pdf/39

16 success of his theoretical investigations, we may refer to the well-known demonstration of the theorem (usually attributed to Laplace) relative to the repulsion exercised by a charged conductor on a point near its surface. This theorem may be stated as follows:-Let A be a closed surface of any form, and let matter, attracting inversely as the square of the distance, be so distributed over it that the resultant attraction on an interior point is nothing: the resultant attraction on an exterior point, indefinitely near any part of the surface, will be perpendicular to the surface and equal to $$4\pi p$$, if $$pw$$ be the quantity of matter on an element $$w$$ of the surface in the neighbourhood of the point. Coulomb's demonstration of this theorem may be found in a preceding paper in the Mathematical Journal, vol. iii. p. 74 (above, 1. 7). He gives it himself, in his sixth memoir on Electricity (Histoire de l'Académie, 1788, p. 677), in connexion with an investigation of the theory of the proof plane in which, by an error that is readily rectified, he arrives at the result that a small insulated conducting disc, put in contact with an electrified conductor at any point, and then removed, carries with it as much electricity as lies on an element of the conductor at that point equal in area to the two faces of the disc; the quantity actually removed being only half of this. This result, however, does not at all affect the experimental use which he makes of the proof plane, which is merely to find the ratios of the intensities at different points of a charged conductor. As the complete theory of this valuable instrument has not, so far as I am aware, been given in any English work, I annex the following remarkably clear account of it, which is extracted from Pouillet's Traité de Physique:-"Quand le plan d'épreuve est tangent à une surface, il se confond avec l'élément qu'il touche, il prend en quelque sorte sa place relativement à l'électricité, ou plutôt il devient luimême l'élément sur lequel la fluide se répand ; ainsi, quand on retire ce plan, on fait la même chose que si l'on avait découpé sur la surface un élément de même épaisseur et de même étendue que lui, et qu'on l'eût enlevé pour le porter dans la balance sans qu'il perdit rien de l'électricité qui le The memoirs of Poisson, on the mathematical theory, contain the analytical determination of the distribution of electricity on two conducting spheres placed near one another, the solution being worked out in numbers in the case of two equal spheres in contact, which had been investigated experimentally by Coulomb (as well as in another case, not examined by Coulomb, which is given as a specimen of the numerical results that may be deduced from the formulæ). The calculated ratios of the intensities at different points of the surface he is therefore enabled to compare with Coulomb's measurements, and he finds an agreement which is quite as close as could be expected, when we consider the excessively difficult and precarious nature of quantitative experiments in electricity: but the most remarkable confirmation of the theory from these researches is the entire agreement of the principal features,