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This is the first part of a two-part paper which summarizes work pursued by the author in 1966 . The paper describes the application of minimum-variance estimation techniques for in-flight alignment and calibration of an inertial measurement unit (IMU) relative to another IMU and/or some other reference. The first part formulates the problem, and the second part  reports numerical results and analyses. The approach taken is to cast the problem into the framework of Kalman-Bucy estimation theory, where velocity and position differences between the two IMU's are used as observations and the IMU parameters of interest become part of the state vector. Instrument quantization and computer roundoff errors are considered as measurement noise, and environmental induced random accelerations are considered as state noise. Typical applications of the technique presented might include the alignment and calibration of IMU's on aircraft carriers, the initialization of rockets or rocket airplanes which are launched from the wing of a mother ship, the alignment and calibration of IMU's which are only used in the latter phases of rocket flight, and for the initialization/updating of SST guidance systems.