Intelligence and Miscellaneous Articles. 459 



with the densities !) x , f) 2 , &c, I), the potentials of which are 



$ v % &c, 33. 

 Kow, if c?m^ i5 cZw 2 , &c, cfov represent the surface-elements of C ]5 C 2 , 

 &c., C, 



jX^K + J V a |j a cZw a + &c. = ^ 1 \div 1 +$$Ji 2 dw 2 + &c, 

 or even 



2 f Y^dw = '2^hdw (1) 



holds good universally. 



Before demonstratiug this theorem, I remark that it can be ap- 

 plied directly to a number of conducting bodies C x , C 2 , &c. charged 

 with electricity in two different ways and acting by influence upon 

 each other ; for then the potentials V and 23 are constants express- 

 ing the potential-levels of the conductors, and equation (1) changes 

 into the equation of Clausius. 



In order to prove the above equality, we start, as Clausius did, 

 from Green's well-known equation. This equation is employed for 

 the space enclosed by the surface C ; and for the two functions 

 occurring therein I take Y t - and 23*, of which the first represents 

 the potential of all the layers with densities h, and the second the 

 potential of the layers with densities 1), both potentials taken in 

 points within C. We then have 



The operation — is the differentiation with respect to the nor- 

 mal drawn within C. By successive applications of Green's theorem 

 to the spaces enclosed by all the surfaces, we get, after summation, 



2k.^=4^'... 00 



Green's proposition is further applied to the space outside the 

 surface C, which space was first bounded by the surface of a sphere; 

 the centre and radius were taken so that all the surfaces C lay within 

 the sphere. The potentials of the layers h and f) in points outside 

 the surfaces C are represented respectively by V a and 23 a , and these 

 functions substituted in Green's equation. "We then find, in the 

 usual way, if the radius of the sphere be infinite, 



2 i T #*- 2 ^S* < 3 > 



The operation -— i s the differentiation with respect to the normal 



drawn outside of the surface C. The functions Yt and Y a in equa- 

 tions (2) and (3) have of course the same value V for the same 

 point of one of the surfaces C ; likewise the functions 23 7 - and 23 a , 

 which both change into 23. It is not so with the differential quo- 

 tients along the normal. 



2K2 



