Mathematical investigations concerning the laws of the equilibrium of fluids analagous to the electric fluid / by George Green.

  • George Green
Date:
1833
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Licence: Public Domain Mark

Credit: Mathematical investigations concerning the laws of the equilibrium of fluids analagous to the electric fluid / by George Green. Source: Wellcome Collection.

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    It is not my chief design in this paper to determine mathematically the density of the electric fluid in bodies under given circumstances, having elsewhere* given some general methods by which this may be effected, and applied these methods to a variety of cases not before submitted to calculation, JVIy present object will be to determine the laws of the equilibrium of an hypothetical fluid analagous to the electric fluid, but of which the law of the repulsion of the particles, instead of being inversely as the square of the distance, shall be inversely as any power n of the distance; and I shall have more particularly in view the determination of the density of this fluid in the interior of con- ducting spheres when in equilibrium, and acted upon by any exterior bodies whatever, though since the general method by which this is effected will be equally applicable to circular plates and ellipsoids. 1 shall present a sketch of these applications also. It is well known that in enquiries of a nature similar to the one about to engage our attention, it is always advantageous to avoid the direct consideration of the various forces acting upon any particle p of the fluid in the system, by introducing a particular function V of the co-ordinates of this particle, from the differentials of which the values of all these forces may be immediately deduced f. We have, therefore, in the present paper endeavoured, in the first place, to find the value of V, where the density of the fluid in the interior of a sphere is given by means of a very simple consideration, which in a great measure obviates the difficulties usually attendant on researches of this kind, have been able to determine the value V, where p, the density of the fluid in any element civ of the sphere’s volume, is equal to the product of two factors, one of which is a very simple function containing an arbitrary exponent /3, and the remaining one f is equal to any rational * Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. t This function in the present case will be obtained by taking the sum of all the molecules of a fluid acting upon p, divided by the (h —1)'*' power of their respective distances fromp; .and indeed the function which Laplace has represented by V in the third book of the Mecanique Celeste, is only a particular value of our more general one produced by writing 2 in the place of the general exponent w.
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