GENERAL PROPERTIES OF A NETWORK OF CONDUCTORS. 339 



currents, for these are still closed circuits ; hence the denominator 

 does not contain the combination in question. 



Thus the common denominator of the equations, resolved in 

 respect of the intensities, contains all the combinations of necessary 

 wires, and them only. 



Finally, all the combinations are of the same sign. For if we 

 suppress / - i necessary wires, the network only contains one 

 closed circuit. The fraction which gives the intensity of the 

 current in this circuit then appears in the form - , but after 

 suppressing a common factor it should give 



All the resistances which form the residual circuit enter the 

 denominator in terms of the same sign; hence all the terms are 

 of the same sign. 



It may be observed that all the conductors which terminate at 

 the same summit do not at the same time form part of a system 

 of necessary wires, for if they are all suppressed except one, it is 

 clear that the latter wire remains open. 



945. Lastly, there is a remarkable correlation between the 

 elements of the two wires of the network. Let r^ and r% be any 

 two wires, e 1 and e 2 the electromotive forces they contain; the 

 corresponding intensities t\ and / 2 will be determined by equations 

 of the form 



N 2 



The numerator is obtained by replacing the factor r 1 in each 

 of the combinations which contains A by the second member 

 of the corresponding equation. This numerator contains then the 

 combinations pi and p i together of the resistances which 

 leave a simple closed circuit of which r^ forms part. 



On the other hand the terms of N\ which contain e 2 result 

 themselves from equations in which r^ enters, and therefore of 

 simple circuits of which r^ forms part. 



Z2 



