On a General Theorem respecting Electrical Influence. 455 



-^2 p^2 pv2 



considered, by A is signified the operation ^-^ + ^—2 + ^2^, 



and the integrals which contain dr are to be extended over 

 the entire space under consideration. 



But now, with both the adopted methods of charging, no 

 electricity is contained in this space ; and consequently there 

 hold good in it everywhere the equations 

 Ay=0 and A^ = 0, 

 by which equation (1) reduces to 



i^f."-^' 



»gj.. . . , . (2) 



In this equation also occurs a simplification. The two inte- 

 grals are, according to the above, to be taken over the surfaces 

 of the given bodies and over the spherical surface. But now, 

 if R signifies the infinite radius of the spherical surface, the 

 values of Y and ^ on that surface are infinitesimal quantities 



of the order p, and the values of the differential coefficients 



dV ^ . . 



^r^ — and :r — at which the direction of the normal coincides 



on on 



with that of the radius are infinitely small quantities of the 



order ^. The products V^— and ^ ;^— are consequently, on 



the spherical surface, infinitesimal quantities of the order 



pg. As for the surface-element of the spherical surface, we 



can replace it by the product l^^da, if da denotes an ele- 

 ment of the solid angle at the centre of the spherical surface. 

 Then the factor, on both sides of the equation, with which da 

 under the integral-symbol is affected is an infinitesimal one of 



the order -p, whence it follows that, of both integrals, the part 



which refers to the spherical surface, and which is to be taken 

 according to a from to 47r, is infinitely small and may be 

 neglected. Consequently the integrals in equation (2) need 

 only to be referred to the surfaces of the given bodies. 



On the surface of each body the potential-function is con- 

 stant ; hence, for the part of the integral which refers to it, it 

 can be taken out of the symbol of the integral. Accordingly 

 we can write equation (2) thus : — 



J on J d» J an 



(3) 



