120 ON THE LAWS OF 



It is not my chief design in this paper to determine mathe- 

 matically 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 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. 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 par- 

 ticular function V of the co-ordinates of this particle, from the 

 differentials of which the values of all these forces may be imme- 

 diately deducedt. We have, therefore, in the present paper 

 endeavoured, in the first place, to find the value of F, 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 



* Essay on the Application of Mathematical Analysis to the Theories of Elec- 

 tricity and Magnetism. 



f Tiiis 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 (w-l) th power of their 

 respective distances from p; and indeed the function which Laplace has repre- 

 sented 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 ex- 

 ponent n. 



