INTRODUCTORY OBSERVATIONS. 



THE object of this Essay is to submit to Mathematical Analysis 

 the phenomena of the equilibrium of the Electric and Magnetic 

 Fluids, and to lay down some general principles equally ap- 

 plicable to perfect and imperfect conductors ; but, before enter- 

 ing upon the calculus, it may not be amiss to give a general 

 idea of the method that has enabled us to arrive at results, 

 remarkable for their simplicity and generality, which it would 

 be very difficult if not impossible to demonstrate in the ordi- 

 nary way. 



It is well known, that nearly all the attractive and repul- 

 sive forces existing in nature are such, that if we consider any 

 material point p, the effect, in a given direction, of all the 

 forces acting upon that point, arising from any system of bodies 

 S under consideration, will be expressed by a partial differential 

 of a certain function of the co-ordinates which serve to define 

 the point's position in space. The consideration of this function 

 is of great importance in many inquiries, and probably there are 

 none in which its utility is more marked than in those about to 

 engage our attention. In the sequel we shall often have occasion 

 to speak of this function, and will therefore, for abridgement, call 

 it the potential function arising from the system S. If p be a 

 particle of positive electricity under the influence of forces arising 

 from any electrified body, the function in question, as is well 

 known, will be obtained by dividing the quantity of electricity 

 in each element of the body, by its distance from the particle p> 

 and taking the total sum of these quotients for the whole body, 

 the quantities of electricity in those elements which are negatively 

 electrified, being regarded as negative. 



