10 INTRODUCTORY OBSERVATIONS. 



It is by considering the relations existing between the density^ 

 of the electricity in any system, and the potential functions 

 thence arising, that we have been enabled to submit many 

 electrical phenomena to calculation /which had hitherto resisted 

 the attempts of analysts ; and the generality of the consideration 

 here employed, ought necessarily, and does, in fact, introduce a 

 great generality into the results obtained from it. There is 

 one ^consideration peculiar to the analysis itself, the nature 

 and utility of which will be best illustrated by the following 

 sketch. 



Suppose it were required to determine the law of the dis- 

 tribution of the electricity on a closed conducting surface A 

 without thickness, when placed under the influence of any 

 electrical forces whatever: these forces, for greater simplicity, 

 being reduced to three^X, Y", and Z^& the direction of the rect- 

 tirgtrhar ^co-ordinates, and tending to increase them. Then p 



V 



CIJ.1 _L IAC*A \SW V**Vfc****W**V*Ji CIJ.J.VL VX/JLfcVUUbU* CV/ AX-LV^i VCIOVJ Cll^li.1* JL. JLL \jlJi L/ 



"*! (jmfepresenting the density of the electricity on an element dcr of 

 L the surface, and r^the distance between dcr and p, any other 

 point of the surface, the equation for determining p which would 

 be employed in the ordinary method, when the problem is re- 

 duced to its simplest form, is known to be 



--* ?. 



Ydy + Zdz] (a); 



the first integral relative to dcr extending over the whole surface 

 A, and the second representing the function whose complete 

 differential is Xdx + Ydy + Zdz, x, y and z being the co-ordinates 



This equation is supposed to subsist, whatever may be the 

 position of />, provided it is situate upon A. But we have no 

 general theory of equations of this description, and whenever we 

 are enabled to resolve one of them, it is because some con- 

 sideration peculiar to the problem renders, in that particular 

 case, the solution comparatively simple, and must be looked 

 upon as the effect of chance, rather than of any regular and 

 scientific procedure. 



We will now take a cursory view of the method it is pro- 

 posed to substitute in the place of the one just mentioned. 



