131] ELECTRICAL EQUILIBRIUM. 257 



for the derivative of V is continuous on crossing the surface, as 

 none of the distribution causing V lies on the surface. Accord- 

 ingly the surface equation becomes 



8(F+ F') 



(16) ^ - ^ = -47T(r. 



dn e 



The surface density at any point of a conductor is the deriva- 

 tive of the total potential in the direction of the external normal 



to the conductor multiplied by T , that is to the field-strength 



directed away from the conductor divided by 4?r. This theorem 

 is due to Coulomb, at least by implication*. The total charge of 

 a conductor K s is 



(17) e e 



The following form of the investigation is shorter, and depends 

 on the variation of the second form of the integral 



where we put, as we shall hereafter do, V for the total potential, 

 heretofore denoted by F + V, 



9F8SF . 



+ 



* tf v &r+m d s- i fff 



4?r J J K { dni dn e j 4?r J J J 



VSadS + VBpdr. 



Now in D, Sp 0, and in external space p = 0, so that the 

 volume integral can be taken through K only, and introducing the 

 equations of condition $e s = 0, 



(19) 2 8 (V-c s ) MS + (F-c s )v =0, 



* Coulomb. "Suite des recherches sur la distribution du fluide electrique 

 entre plusieurs corps conducteurs. " 1788. Collection de mem. rel. ct la physique, 

 pub. par la Soc. frang. de Phys. Tom. i. p. 230. 



W. E. 17 



