360] General Theory 315 



the outset. Thus we arrive at the conclusion that the system of equations 

 (291) are not independent. 



This is as it should be, for if the equations were independent, we should have 

 n equations from which it would be possible to determine the values of F 1} F 2 , ... in 

 terms of J^, JT 2 , ... ; whereas clearly from a knowledge of the currents entering the 

 network, we must be able to determine differences of potential only, and not absolute 

 values. 



To the right-hand side of equation (291), let us add the expression 



of which the value is zero by the definition of K u . The equation becomes 



#u (K - V n ) + K a (TJ - V n ) + . . . + #i,n-i ( JU - V n ) 



= X j + ^12^12 + KisElS 4- ... + K ln E ln . 



There are n equations of this type in all. Of these the first (nl) may 

 be regarded as a system of equations determining 



V V V V V V 



'i 'n> "2 MJ > "ni~ "n- 



That these equations are independent will be seen a posteriori from the fact 

 that they enable us to determine the values of the n 1 independent 

 quantities 



V V V V V V 



'I 'n> "z *m > 'ni "n- 



Solving these equations, we have 



X ! + K 12 E 12 4- ... + ^m^m, JT||| 



"hi *^M) ^13; > J*-i,n l 



-**- 21 J -** 22 > -"~23> ) " 2, nl 



"-fcfl, -***n i,2j **-n 1,3> ) &n i,n 



The current flowing in conductor In follows at once from equation (289), 

 and the currents in the other conductors can be written down from sym- 

 metry. 



If we denote the determinant in the denominator of the foregoing 

 equation by A, and the minor of the term K PQ by A PQ , we find that the 

 value of V l V n can be expressed in the form 



V t -V n = (-X l + K K E K + ...+K ln E m )^ 



a + ... + K w E w ) + (292). 



