OP AN ELECTRIC DISCHARGE. 9 



In the first place, let it be required to determine the value of 

 the potential in the case of a charged Leyden jar or battery. 



In order to present the matter more simply to the mind, we 

 will commence with the special case of a jar of particular form, 

 and apply the expression thus found to the deduction of the 

 general expression. .We will first choose a form without indeed 

 an actual existence, but which, however, in all essential points 

 must be subject to the same laws as the common Leyden jar, 

 and which leads to results of extraordinary simplicity. The 

 glass vessel shall form a closed hollow sphere, possessing at all 

 points the same thickness, and completely covered with tinfoil 

 both inside and out. Let us suppose a quantity of electricity 

 Q to be imparted by some means or other to the interior sur- 

 face, where we assume as unit such a quantity of positive elec- 

 tricity as exerts upon an equal quantity of the same electricity 

 the unit of force at the unit of distance. Let the outer surface 

 stand in connexion with the earth, and let the quantity of elec- 

 tricity which it receives from the latter be denoted by QJ, 



In this case it is evident that Q and Q! must spread them- 

 selves uniformly over both the respective surfaces, and in con- 

 sequence of this the determination of the potential function and 

 of the potential is greatly facilitated. 



The potential function V of any quantity of electricity Qi 

 at any point O, will, in general, be determined by the equation 



*dq 



^'-f- 



v 



where dq denotes an element of electricity, and r its distance 

 from the point O, the integral extending over the entire quan- 

 tity. For the particular case however where Ql is spread uni- 

 formly over a spherical surface we do not need this general 

 equation, but can apply the following two propositions: — 



1. Within the sphere the potential function is everywhere the 

 same, that is, if r be the radius of the sphere, 



r 



2. Without the sphere, and for a distance R from its centre, the 

 potential function is 



V — « 

 R* 



