240 Dr. J. A. Fleming on the Distribution of 



each equated to the effective electromotive force in that cycle 

 or mesh. 



Let us assume now that x is constant, but that y, z, u, w, &c. 

 are independent variables and are arbitrarily changed. This 

 is equivalent to supposing that a given quantity of electricity 

 per second is pushed into the network, but that its distribution 

 is supposed to be varied. We see that the equations which 

 we write down, according to Maxwell, to determine the real 

 distribution of currents in the network, according to Ohm's 

 law, are the same equations as would be written down to find 

 the values of y, z, u, w, &c, which make the dissipation func- 

 tion a minimum under fixed conditions of total current flow- 

 ing into the network, viz. equating to zero the first partial 

 differentials of H with respect to the variables y, z, u, &c. 

 The same holds good generally, hence we see that this is 

 another way of arriving at the theorem of which Maxwell has 

 given a proof on page 375, § 284, vol. i. of his large Treatise, 

 2nd edition, viz.: — " In any system of conductors in which 

 there are no internal electromotive forces the heat generated 

 by currents distributed in accordance with Ohm's law is less 

 than if the currents had been distributed in any other manner 

 consistent with the actual conditions of supply and outflow of 

 the current." 



The exact proof that the partial differentials of the dissipa- 

 tion function equated to zero gives the condition that the dis- 

 sipation function shall be a minimum is not complete without 

 an examination of Lagrange's conditions. It is obvious that 

 the second partial differentials of the dissipation function are 

 quantities which are resistances, viz. the coefficients of the 

 current symbols in the cycle equations, and that the conditions 



for a minimum are complied with, since ^TTj ^ c * are P 08 *" 



tive ; and the discriminant of the quadratic function of the 

 currents or symmetrical determinants formed of these second 

 partial differentials is what has been called above the network 

 determinant. This and all its successive minors are positive 

 quantities*. 



§ 14. In the foregoing sections the problems have been 

 treated under the limitations that the various meshes of the 

 network of conductors have no mutual and no self-induction. 

 The introduction of these inductive actions will affect in a 

 considerable way the treatment of the problem ; and the dis- 



* See Williamson's l Differential Calculus/ p. 408, " On the Conditions 

 for a Maximum and Minimum of a Function of any number of Variables," 

 § 163, and Appendix. 



