450 A new Electrical Theorem. 



For it is plain from the method of derivation, that if a seat of 

 electromotive force exists in any closed Kirchhoff path it can 

 never leave it ; and if in the movement of the sources one of 

 them approaches the closed path under consideration, at the 

 encounter it becomes in that path two equal sources acting in 

 opposite directions. 



If, therefore, %e remains the same for any path and r remains 

 the same for every part, then obviously c must remain the 

 same for any portion of that path, and therefore for every 

 part of the network. 



The following propositions flow immediately from the main 

 proposition : — 



(1) If a closed continuous surface contains any region of 

 any network, and some bar which cuts the surface contains, or 

 by derivation as above can be made to contain, the seat of an 

 electromotive force, then that source can be done away with 

 without disturbing the currents in any portion of the system 

 provided that in the other bars which cut the surface sources 

 of electromotive force be inserted of equal value but of 

 opposite direction as regards inside and outside of the sur- 

 face ; for it is clear that such sources would result from the 

 migration of sources in one direction. 



(2) If two systems of electromotive forces are equivalent, 

 one may be derived from the other. For if system A is equi- 



A 



valent to system B, and we suppose -^ to represent a distribu- 

 tion identical with A as regards positions, but of half the 



A B 



electromotive force in any case, then — + — is equivalent to 



A or B alone. Now if any Kirchhoff path containing a 



A B 



source from -r- does not also contain a source from -^-, then 



Kirchhoff's law would be outraged ; for the sum of the electro- 

 motive forces in that path would be only half what they are 

 from A alone, whereas the currents and resistances remain 

 the same. Hence for every Kirchhoff path there must be 

 equal sources from each system. Either system may now 

 have its elements moved up to those of the other system ; 

 any resulting side branchings will be the same (though dif- 



A B 



fering in sign) whether derived from -^ or from ^-, and 



— — 



must necessarily produce no effect by themselves, because if 

 we consider -^ to be reversed, the whole effect of -^ and -^, 

 now in the same bars, will be zero, 



