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A marching-on-in-degree (MOD) based time-domain magnetic field integral equation method for bodies of revolution (BOR) is proposed and applied to obtain the induced currents on perfectly electric conducting BOR. Before this work, the time-domain integral equation method for BOR based on a marching-on-in-time procedure cannot really reduce the computational cost, since the number of unknowns cannot really be reduced. But it is the unknown reduction that serves as the key point of cost saving in BOR-problems. The method implemented in this letter can really utilize the symmetric property of BOR by applying two sets of entire domain basis functions. One is a set of scaled Laguerre polynomials inherited from common MOD method and used as temporal basis functions. The other is a Fourier series which comes from frequency domain method for solving BOR-problems. The validity, efficiency, and stability of the method are verified by several numerical examples.