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In this paper we describe a mechanical procedure for computing the Liapunov functions and Liapunov constants for a class of differential systems. These functions and constants are used for establishing the stability criteria, the conditions for the existence of a center and for the investigation of limit cycles. Some problems for handling the computed constants, which are usually large polynomials in terms of the coefficients of the differential system, and an approach towards their solution by using computer algebraic methods are proposed. This approach has been successfully applied to check some known results mechanically. The author has implemented a system DEMS on an HP1000 and in Scratchpad II on an IBM4341 for computing and manipulating the Liapunov functions and Liapunov constants. As examples, two particular cubic systems are discussed in detail. The explicit algebraic relations between the computed Liapunov constants and the conditions given by Saharnikov are established, which leads to a rediscovery of the incompleteness of his conditions. A class of cubic systems with 6-tuple focus is presented to demonstrate the feasibility of our approach for finding systems with higher multiple focus.