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The epoch-marking publication of Kirchhoff in 1847 is considered in this paper. The rule given by Kirchhoff deserves to be understood by almost all engineers, for it shows how to determine in a simple and straightforward way the system functions of any passive network that contains no transformers. Preliminary to the consideration of the rule and its dual, the basic concepts of combinatorial topology are summarized in the present paper. The rules are then stated in the detail necessary to show that their application can be carried out in a routine and mechanical manner. Application of the rules to problems in analysis and synthesis is illustrated in a number of examples. Included among these examples are the derivation of the necessary and sufficient conditions on RLC networks that have been previously given by Fialkow and Gerst, the analysis of an infinite graph, and the determination of the elements of the chain matrix (or the open-circuit impedance or short-circuit admittance matrix) by essentially a single calculation. Finally, the Feussner method for simplifying the use of the rules is discussed.