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LAWS OF THE EQUILIBRIUM QF FLUIDS

ANALOGOUS TO THE ELECTRIC FLUID,

WITH

OTHER SIMILAR RESEARCHES'

From the Trannadione of the Cambridge Philotophical Society, 1833.
[Read Nov. 12, 1832.]

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MATHEMATICAL INVESTIGATIONS CONCERNING THE LAWS OF THE EQUILIBRIUM OF FLUIDS ANALOGOUS TO THE ELECTRIC FLUID, WITH OTHER SIMILAR RESEARCHES.

Amongst the various subjects which have at different times occupied the attention of Mathematicians, there are probably few more interesting in themselves, or which offer greater difficulties in their investigation, than those in which it is required to determine mathematically the laws of the equilibrium or motion of a system composed of an infinite number of free particles all acting upon each other mutually, and according to some given law. When we conceive, moreover, the law of the mutual action of the panicles to be such that the forces which emanate from them may become insensible at sensible distances, the researches to which the consideration of these forces lead will be greatly simplified by the limitation thus introduced, and may be regarded as forming a class distinct from the rest. Indeed they then for the most part terminate in the resolution of equations between the values of certain functions at any point taken at will in the interior of the system, and the values of the partial differentials of these functions at the same point. When on the contrary the forces in question continue sensible at every finite distance, the researches dependent upon them become far more complicated, and often require all the resources of the modern analysis for their successful prosecution. It would be easy so to exhibit the theories of the equilibrium and motion of ordinary fluids, as to offer instances of researches appertaining to the former class, whilst the mathematical investigations to which the theories of Electricity and Magnetism have given rise may be considered as interesting examples of such as belong to the Utter class.

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. My present object will be to determine the laws of the equilibrium of an hypothetical fluid analogous 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 conducting 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. I 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+. 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 dv 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 /?, and the remaining one / is equal to any rational and entire function whatever of the rectangular co-ordinates of the element dv, and afterwards by a proper determination of the exponent /8, have reduced the resulting quantity Vta a rational and entire function of the rectangular co-ordinates of the particle p, of the same degree as the function f. This being done, it is easy to perceive that the resolution of the inverse problem may readily be effected, because the coefficients of the required factor / will then be determined from the given coefficients of the rational and entire function V, by means of linear algebraic equations.

* Essay on the Application of Mathematical. Analyst* 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 (n-l)th power of their respective distance* from p; and indeed the function which Laplace has represented by V in the third book of the Mtcanique Colette, is only a particular value of our more general one produced by writing 2 in the place of the general exponent n.

The method alluded-to in what precedes, and which is exposed in the two first articles of the following paper, will enable us to assign generally the value of the induced density p for any ellipsoid, whatever its axes may be, provided the inducing forces are given explicitly in functions of the co-ordinates of p; but when by. supposing these axes equal we reduce the ellipsoid to a sphere, it is natural to expect that as the form of the solid has become more simple, a corresponding degree of simplicity will be introduced into the results; and accordingly, as will be seen in the fourth and fifth articles, the complete solutions both of the direct and inverse problems, considered under their most general point of view, are such that the required quantities are there always expressed by simple and explicit functions of the known ones, independent of the resolution of any equations whatever.

The first five articles of the present paper being entirely analytical, serve to exhibit the relations which exist between the density p of our hypothetical fluid, and its dependent function V; but in the following ones our principal object has been to point out some particular applications of these general relations.

In the seventh article, for example, the law of the density of our fluid when in equilibrium in the interior of a conductory sphere, has been investigated, and the analytical value of p there found admits of the following simple enunciation.

The density p of free fluid at any point p within a conducting sphere A, of which 0 is the centre, is always proportional to the In _ 4)u» power of the radius of the circle formed by the inter

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