OF AN ELECTRIC DISCHARGE. 11 



of the entire electricity upon itself. The potential W of a given 

 quantity of electricity upon itself is, in general, 



where dq and d(i represent any two elements of the electricity, 

 r their distance asunder, and both the integrals extend over the 



entire quantity, the factor - must be introduced, because in the 



double integral every combination of every two elements dq and 

 d(^ appears twice. Now as 



-/ 



-*_v, 



r 

 instead of the foregoing expression we can write 



y^=\fydq (8) 



Now, as before mentioned, in every connected conducting body 

 the potential function is constant, and may therefore be taken 

 from under the sign of integration ; the integral that remains 

 represents simply the quantity of electricity distributed over the 

 body. If we apply this to the two coatings of a Ley den jar with 

 the potential functions V and V, we obtain as the total potential 

 of both the quantities Ql and Q! upon themselves, 



W=l(Q.V + a'.V); (9) 



and if for our special case we set V'=0, and in the place of V 

 set the value of it, as found by (7) or (7«)^ we obtain the re- 

 quired potential in the case of a charged spherical jar, 



W=-Q^— £ ,, (10) 



or 



W=-^^27^cfl-f-f ^-&c.V . . (10«) 

 S \ a a^ J 



The plate of Franklin with circular coatings may be regarded 

 as the next simplest form of the Leyden jar. In a former me- 

 moir I have devoted especial attention to this form, and from 

 the results there obtained I will introduce but one here, which 

 corresponds to the foregoing example. This is the case in which 

 one of the metallic coverings is supposed to be connected with 



