PREFACE. 5 



impossible to give any idea of them here : to be duly appretiated 

 they must be read. It will therefore only be remarked, that 

 they are in fact founded upon the consideration of what have, 

 in this Essay, been termed potential functions, and by means 

 of an equation in variable differences, which may immediately 

 be obtained from the one given in our tenth article, serving to 

 express the relation between the two potential functions arising 

 from any spherical surface, the author deduces the values ot 

 these functions belonging to each of the two spheres under con- 

 sideration, and thence the general expression of the electric 

 density on the surface of either, together with their actions on 

 any exterior point. 



I am not aware of any material accessions to the theory of 

 electricity, strictly so called, except those before noticed; but 

 since the electric and magnetic fluids are subject to one common 

 law of action, and their theory, considered in a mathematical 

 point of view, consists merely in developing the consequences 

 which flow from this law, modified only by considerations arising 

 from the peculiar constitution of natural bodies with respect to 

 these two kinds of fluid, it is evident the mathematical theory 

 of the latter, must be very intimately connected with that of the 

 former; nevertheless, because it is here necessary to consider 

 bodies as formed of an immense number of insulated particles, 

 all acting upon each other mutually, it is easy to conceive that 

 superior difficulties must, on this account, present themselves, 

 and indeed, until within the last four or five years, no successful 

 attempt to overcome them had been published. For this farther 

 extension of the domain of analysis, we are again indebted to 

 M. PoiSSON, who has already furnished us with three memoirs 

 on magnetism: the first two contain the general equations on 

 which the magnetic state of a body depends, whatever may be 

 its form, together with their complete solution in case the body 

 under consideration is a hollow spherical shell, of uniform thick- 

 ness, acted upon by any exterior forces, and also when it is a 

 solid ellipsoid subject to the influence of the earth's action. By 

 supposing magnetic changes to require time, although an ex- 

 ceedingly short one, to complete them, it had been suggested 

 that M. ARAGO'S discovery relative to the magnetic effects 



