COMPARISON OF RESISTANCES. 



the addition of the conductor r brings into the system a new 

 equation, and one only. 



When a system of p necessary wires is suppressed, the network 

 is entirely open and cannot give rise to any equation of the latter 

 form. The successive addition of p necessary wires, which re- 

 produces the original network, introduces p distinct equations, 

 which demonstrates the proposition. 



As the network contains n different conductors, and the 

 physical phenomenon is denned, the sum total of the equations 

 should be equal to the sum n of the intensities to be determined; 

 from this follows the condition 



p + m- i=n, 

 or 



p n-m+ i . 



The least number p of necessary conductors is thus denned by 

 the number of summits, and the number of sides of the network. 



944. The intensities being multiplied by their respective 

 resistances in the / equations (19) for the closed circuits, and by 

 i in the m-i equations (18) for the summits, the common 

 denominator A determined by the ordinary rule, comprises the 

 combinations p and / together of the different resistances. 



We shall obtain, moreover, the numerator of the fractions which 

 expresses the value of the intensity i k , if we replace in trie de- 

 nominator A the coefficient r k of t k in each of the equations by 

 the known corresponding term. The numerator contains these 

 electromotive forces multiplied respectively only by sums of the 

 combinations p - i and pi together of the resistances. 



Every combination r^ , r z . . . r p of p necessary wires enters into 

 the denominator. For when the resistances are made infinite, 

 which is equivalent to suppressing the corresponding wires, there 

 is no closed circuit, and the equations should be satisfied if the 

 values for the intensities are zero. But if we divide the two 

 terms of the fraction which gives an intensity /", by the product 

 r\, r 2 . . . r p the numerator is null because it only contains com- 

 binations of resistances p - i and p-i together ; as the denomi- 

 nator cannot be null, it must contain the combination r^ r^ . . . r p . 



Conversely, if a combination r lt r 2 . . . r p does not correspond to 

 a system of necessary wires, and we repeat the same reasoning, 



the fraction should appear in the form of -- for some of the 



