The extremal length of a plane region is defined by a minimax type limit of certain geometric quantities. Extremal length is invariant under conformal mapping and in this light it has been studied by Gr6tzsch, Beurling, and Ahlfors [1]. In this paper the definition of extremal length is extended to discrete systems. This permits defining the extremal length (and extremal width) between any two nodes of an electric network.
It is then proved that the extremal length between two nodes of a network is identical with the joint resistance between these nodes. The method of proof employs linear programming theory. In particular the max-flow equals rain-cut theorem of Ford and Fulkerson [2] is used. In addition another theorem of this type is needed; we term it "max-potential equals min-work."
In a previous paper a planar network was considered and a dual associated network termed the "conjugate" was defined . It was found that the joint resistance of the conjugate network is the reciprocal of the joint resistance of the original network. A simple proof of this result is given here by making use of the method of extremal length.
The last section of the paper gives an inequality relating paths and cuts in a network. This inequality results from the theorem that extremal width and extremal length are reciprocals.
II. RAYLEIGH'S RECIPROCAL THEOREM
Consider the problem of finding the electrical resistance of a plate in the shape of a curvilinear quadrilateral as shown in Fig. . The resistance p is required between edges A and B. The edges C and D are supposed insulated. The plate is of unit thickness and the top and bottom are insulated.