MATHEMATICAL INVESTIGATIONS CONCERNING THE 

 LAWS OF THE EQUILIBRIUM OF FLUIDS ANA- 

 LOGOUS 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 diffi- 

 culties 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 particles 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 latter class. 



