For ideal, general first-order nonholonomic constrained systems, this paper argues that the relation satisfied by the virtual displacement, namely Chetaev relation, can be derived, but the virtual displacement in the relation should refer to the Gauss variation. Similarly, in order to derive the Lagrange equation with multipliers from the general equations of dynamics using Chetaev relations, it is necessary to first determine whether the system is an ideal constrained system, and the virtual displacement in the ideal constraint conditions should also refer to the Gauss variation. For an ideal, first-order linear nonholonomic constrained system, in the relation and ideal constraint conditions satisfied by the virtual displacement, the virtual displacement can be either Gauss variation or Jourdain variation.