2 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



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 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 F^, 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 fi, and the remaining one J" 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 (n — 1)* power of their respective distances from^; 

 and indeed the function which Laplace has represented by F 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 n. 



