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In this paper, we present the full derivation of a new formula for calculating the mutual inductance between inclined circular filaments arbitrarily positioned with respect to each other. Although such a formula was already proposed by Grover more than 50 years ago, the formula presented here is slightly more general and simpler to use, i.e., it involves only a sequential evaluation of expressions and the numerical resolution of a simple numerical integration. We derived the new formula using the method of vector potential, as opposed to Grover's approach, which was based on the Neumann formula. We validated the new formula through a series of examples, which are presented here. Finally, we present the relationship between the two general formulas (i.e., Grover's and our new one) explicitly (Example 12).