TO THE THEORY OF ELECTRICITY. 51 



mentioned, that the potential function V, arising from the same 

 spherical surface, and belonging to a point p, exterior to this 

 surface, at the distance r from its centre, and on the radius r 

 produced, will be 



r = Z7 (0) % + U (l) ~ + Z7 (2) + etc. 



If, therefore, we make V=<t> (r), and V ^ (/), the two func- 

 tions <f) and -|r will satisfy the equation 



* r 



But (art. 4) 



dv dV dv 



and the equation between < and -v/r, in its first form, gives, by 

 differentiation, 



Making now r = a there arises 



^>' and ty' being the characteristics of the differential co-efficients 

 of </> and o/r, according to Lagrange's notation. 



In the same way the equation in its second form yields 



These substituted successively^ in the equation by which p 

 is determined, we have the following, 



dr a 



dr a 



(9). 



If, therefore, the value of the potential function be known, 

 either for the space within the surface, or for that without it, the 



42 



