Abstract: Rayleigh–Kuo and Fjørtoft theorems provide the necessary conditions for barotropic instability of a geophysical fluid flow. This instability corresponds to a geostrophic flow is crucial for the geophysical flow; therefore, this paper modifies two necessary conditions for this instability. The implication of the necessary conditions for barotropic instability is explained based on the energetics and potential vorticity dynamic theories. Rayleigh–Kuo theorem for the geostrophic flow is revised as follows: As a necessary condition for the barotropic instability of the geostrophic flow, the potential flow vorticity should have an extreme point. The Fjørtoft theorem (Fjørtoft, 1950) correspondingly changes to the following: As a necessary condition for the barotropic instability of the geostrophic flow, Q_y (U−U_s)>0 in the geostrophic flow field, where y_s is a point at which Q_y=0 and U(y_s)=U_s. These modified conditions indicate that the absolute vorticity in the original theorems is modified to potential vorticity, which is a consequence of the geostrophic and potential vorticity constraints. The energy relation exhibits the energetics requirement for the potential barotropic instability. Rayleigh–Kuo theorem is a potential vorticity dynamic condition for the possible barotropic instability, and the Fjørtoft theorem is the coordination of two requirements corresponding to energetics and potential vorticity dynamics. All these three requirements must be met for the barotropic instability.