183-187] Uniqueness of Solution 161 



Greens Reciprocation Theorem. 



185. In equation (101), put <!>= F, = F', where F is the potential 

 of one distribution of electricity, and V is that of a second and independent, 

 distribution. The equation becomes 



which is simply the theorem of 102, namely 



2eF'=2e'F .............................. (104). 



If we assign the same values to <E>, M* in equation (103), we again obtain 

 equation (104), which is now seen to be applicable when dielectrics are 

 present. 



UNIQUENESS OF SOLUTION. 



186. We can use Green's Theorem to obtain analytical proofs of the 

 theorems already given in 99. 



THEOREM. If the value of the potential V is known at every point on 

 a number of closed surfaces by which a space is bounded internally and 

 externally, there is only one value for V at every point of this intervening 

 space, which satisfies the condition that V 2 F either vanishes or has an assigned 

 value, at every point of this space. 



For, if possible, let F, V denote two values of the potential, both of which 

 satisfy the requisite conditions. Then F' F=0 at every point of the 

 surfaces, and V 2 (F' F) = at every point of the space. Putting <I> and M^ 

 each equal to V Fin equation (100), we obtain 



and this integral, being a sum of squares, can only vanish through the 

 vanishing of each term. We must therefore have 



( r-y) = <r'-F)=(F'-F) = ............ (105 X 



or V V equal to a constant. And since V V vanishes at the surfaces^ 

 this constant must be zero, so that V=V everywhere, i.e. the two solutions 

 F and V are identical : there is only one solution. 



187. THEOREM. Given the value of ^ at every point of a number of 



on 



closed surfaces, there is only one possible value for V (except for additive 

 constants), at each point of the intervening space, subject to the condition that 

 V 2 F = throughout this space, or has an assigned value at each point. 



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