ABSOLUTE formulas for the mutual inductance of coaxial solenoids have been derived by Nagaoka, Terezawa, and Olshausen. These all give the mutual inductance as a combination of 4 integrals — 2 positive and 2 negative — each of which is a function of one of the 4 distances measured between an end of one solenoid and an end of the other. The expressions for these terms involve elliptic integrals of all 3 kinds, and numerical computations of their values are exacting.
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Dwight et al. (1937) studied this question.
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