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Chemical dissolution-front instability (CDFI) problems usually involve multiple temporal and spatial scales, as well as multiple processes. A key issue associated with solving a CDFI problem in a fluid-saturated rock is to mathematically establish a theoretical criterion, which can be used to judge the instability of a chemical dissolution-front (CDF) propagating in the fluid-saturated rock. This theoretical paper deals with how two different mathematical schemes can be used to precisely establish such a theoretical criterion in a purely mathematical manner, rather than in a numerical simulation manner. The main distinguishment between these two different mathematical schemes is that in the first mathematical scheme, a curved surface coordinate system is used, while in the second mathematical scheme, a planar surface coordinate system is employed. In particular, all the key mathematical deduction steps associated with using these two different mathematical schemes are described and discussed in great detail. The main theoretical outcomes of this study have demonstrated that (1) two different mathematical schemes under consideration can produce exactly the same theoretical criterion; (2) the main advantage of using the first mathematical scheme is that the interface conditions at the curved interface between the downstream and upstream regions can be easily described mathematically; (3) the main advantage of using the second mathematical scheme is that the first-order perturbation equations of the CDFI problem can be easily described in a purely mathematical manner.