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LXI. On a General Theorem respecting Electrical Influence. 

 By Professor R. Clausius*. 



CERTAIN reciprocity-theorems respecting the reciprocal 

 influence of two conductors of electricity have already 

 been repeatedly advanced by various authors. I take leave 

 to communicate here a very general, and, so far as I know, 

 new theorem, from which several of those theorems result as 

 immediate consequences. 



Given any number whatever of conducting hoclies Cj, C2, C3, 

 ^c, which act by way of influence on each other. These are to 

 be charged with electricity i7i two different ways. In the first 

 charge let the quantities of electricity present upon the individual 

 bodies be 



Qij Q2J Qsj &c., 

 and the levels of potential thereby produced on the bodies 



Vi, V„ V3, &0.; 

 and in the second charge let the quantities of electricity and the 

 potential-levels be 



Oi, O2, O3, &c., 



«1, ^2, 5^3, &C. 



TJien the following equation holds good: — 

 ViOi + V2O2 + V3D3 + &c. =^iQi + ^2Q2 + ^sQs + &c. (I.) 

 or, employi7ig summation-symbols, more briefly written, 



SV0=2^0 (I A.) 



For the proof of this equation, let us imagine an infinitely 

 large spherical surface formed round a point situated in the 

 vicinity of the bodies, and to the infinite space lying between 

 the bodies and the spherical surface apply the well-known 

 equation of Green, while we denote the two functions therein 

 occurring by V and ^, understanding by these the potential- 

 functions corresponding to the first and second charges respec- 

 tively. The equation will then read : — 



|'v|^^a)+rVA^^T=r^|^6/a>-|-f35AVe^T. . (1) 



Here d(ji signifies an element of the surface which bounds the 

 space we are considering, and which consists of the surfaces 

 of the given bodies and the infinitely large surface of the 

 sphere ; n is to represent the normal erected upon the surface- 

 element (reckoned positive in the direction of the space con- 

 sidered) ; and the integrals containing d(Mi refer to the total 

 limiting surface. Futher, dT denotes an element of the space 



* Translated from a separate impression, communicated by the Author, 

 from roggendorft''s Annalen, new series, vol. i. pp. 493-499. 



