Abstract
Results obtained previously [1, 2], which are applicable to mechanical systems containing non-conservative positional forces, are developed and generalized. The necessary and sufficient conditions are formulated for the transition to a certain matrix equation, the use of which enables one to overcome the difficulties associated with the existence of non-conservative positional structures in the initial equations. The above-mentioned conditions are expressed directly in terms of the matrix coefficients of the initial equation. This technique is used to analyse the exact equations of a four-gyroscope vertical (without using the equations of precessional theory) under the assumption that it is mounted on a base which moves with respect to the Earth.
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