Generalized scaling laws for the formation of plasma confining potentials and the associated effectiveness of the potentials produced are systematically investigated to find the physics essentials common to the representative tandem mirror operational modes of GAMMA 10, and to explore novel extended operational modes from the scaling bases constructed. (a) The potential formation scalings are generalized using a novel finding of wider validity of Cohen's strong ECH theory covering the representative modes. (b) The potentials produced, in turn, provide a favourable novel scaling of the increase in the central cell electron temperatures Te with increasing thermal barrier potentials φ b , limited by the available ECH power. The scaling of Te with φ b is well interpreted in terms of the generalized Pastukhov theory of plasma potential confinement. A detailed comparison of the results from several related modified theories is also made. (c) Consolidation of the two major scalings of (a) and (b) in a tandem mirror is carried out by the use of an electron energy balance equation for the first time. In addition, (d) an empirical scaling of φc with ECH power in the plug region and the central cell densities are studied to discover whether there is the possibility of extending these theoretically well interpreted scaling data to parameters in the future scalable regime. There is also a discussion about numerical scalings in the three dimensional parameter spaces.