4 PREFACE. 



Little appears to have been effected in the mathematical 

 theory of electricity, except immediate deductions from known 

 formula, that first presented themselves in researches on the 

 figure of the earth, of which the principal are, the determina- 

 tion of the law of the electric density on the surfaces of conduct- 

 ing bodies differing little from a sphere, and on those of ellip- 

 soids, from 1771, the date of CAVENDISH'S paper, until about 

 1812, when M. PoiSSON presented to the French Institute two 

 memoirs of singular elegance, relative to the distribution of 

 electricity on the surfaces of conducting spheres, previously 

 electrified and put in presence of each other. It would be quite 



there employed. The object of this proposition is to show, that when two similar 

 conducting bodies communicate by means of a long slender canal, and are charged 

 with electricity, the respective quantities of redundant fluid contained in them, 

 will be proportional to the n - 1 power of their corresponding diameters : sup- 

 posing the electric repulsion to vary inversely as the n power of the distance. This 

 is proved by considering the canal as cylindrical, and filled with incompressible 

 fluid of uniform density : then the quantities of electricity in the interior of the 

 two bodies are determined by a very simple geometrical construction, so that the 

 total action exerted on the whole canal by one of them, shall exactly balance that 

 arising from the other ; and from some remarks in the 2 7 th proposition, it appears 

 the results thus obtained, agree very well with experiments in which real canals 

 are employed, whether they are straight or crooked, provided, as has since been 

 shown by COULOMB, n is equal to two. The author however confesses he is by no 

 means able to demonstrate this, although, as we shall see immediately, it may very 

 easily be deduced from the propositions contained in this paper. 



For this purpose, let us conceive an incompressible fluid of uniform density, 

 whose particles do not act on each other, but which are subject to the same actions 

 from all the electricity in their vicinity, as real electric fluid of like density would 

 be ; then supposing an infinitely thin canal of this hypothetical fluid, whose per- 

 pendicular sections are all equal and similar, to pass from a point a on the surface 

 of one of the bodies, through a portion of its mass, along the interior of the real 

 canal, and through a part of the other body, so as to reach a point A on its sur- 

 face, and then proceed from A to a in a right line, forming thus a closed circuit, it 

 is evident from the principles of hydrostatics, and may be proved from our author's 

 23 d proposition, that the whole of the hypothetical canal will be in equilibrium, 

 and as every particle of the portion contained within the system is necessarily so, 

 the rectilinear portion aA must therefore be in equilibrium. This simple conside- 

 ration serves to complete CAVENDISH'S demonstration, whatever may be the form or 

 thickness of the real canal, provided the quantity of electricity in it is very small 

 compared with that contained in the bodies. An analogous application of it will 

 render the demonstration of the 22 d proposition complete, when the two coatings 

 of the glass plate communicate with their respective conducting bodies, by fine 

 metallic wires of any form. 



