Power-law ( ) and exponential power-law ( ) functional forms model activity metabolism ( ) for fully submerged swimming animals, and are special cases of the power-law polynomial equation,

in which   is the observed total metabolic rate measured at an observed steady swimming speed,  . The relationship between the metabolic efficiency of steady swimming and the exponents of   is addressed in this paper to establish the use of   () and   () as optimal efficiencies for comparing the hydrodynamic and muscle metabolic efficiencies among fully submerged animals that engage in steady swimming activities. The metabolic efficiency of steady swimming is transformed into its ideal form ( ) from which    the optimal hydrodynamic efficiency ( ) and the optimal muscle metabolic efficiency ( ) are derived. These optimal efficiencies are therefore ideal metabolic efficiencies measured at different optimal steady speeds. Subsequently, linear ( ) and exponential ( ) models are approximations with divergent optimal muscle metabolic efficiencies ( and , respectively), but with a similar optimal hydrodynamic efficiency ( ).