456 Prof. R. Clausius on a General Theorem 



in which dxoi^ do>2, dco^, &c. will be surface-elements of the 

 bodies Ci, C2, C3, &c., and the different integrals are to be re- 

 ferred to the surfaces of the respective bodies. 



Further, it is known that a simple relation subsists^ on the 

 surface of a conducting body charged with electricity, between 

 the differential coefficient, taken in the direction of the normal^ 

 of the potential-function, and the electrical density. If, namely^ 

 h and i) denote the surface-densities with the two charges, the 

 equations 



^^— = •— 47r/i and :r — = ~47rb 

 O^ on 



hold, and thereby (3) is transformed into 



Yi^i)day^ + Y2^i)dco2-\-Ys^^dco, + &c.^ 



= ^^]hdoy^ + B2] hdco2 + ^z^ hdo)^ + &G.j 



The integrals occurring here again are, however, nothing 

 more than the quantities of electricity present upon the respec- 

 tive bodies ; and consequently we obtain the equation which 

 was to be demonstrated : — 



YiDi + Y2O2 + V3O3 + &c. = «iQi + ^2Q2 + ^aQs + &c. 



Now the applications of this equation are facilitated by the 

 fact that it can be still further simplified under certain often 

 occurring conditions. 



If, namely, we consider the terms which refer to any one of 

 the given bodies, which may be called C^^, viz. the two products 



Yi£ii and iB,Q^, 



these become in two cases nil, so that they can be omitted 

 from the equation. If the body is in conducting connexion 

 with the earth, its potential-level remains at zero with every 

 charge of the system ; consequently we have to put for this 

 case 



through which the above products vanish. Further, when the 

 body is insulated and initially unelectric, and receives, on 

 charging, no electricity from without, but only through influ- 

 ence undergoes an unequal distribution of its own electri- 

 city, then its surface becomes in part positively, and in part 

 negatively electric, in such wise that the total electricity pre- 

 sent on the surface remains nil. We have consequently then 

 to put 



Q-O,=o, 



through which the above products again vanish. Accordingly 



