We derived the general forms of the basic zoom equations of the mechanical compensation type system by using Gaussian brackets' techniques. We found that the zoom systems are distinguished into S-type system that first movable member is variator and T-type system that second movable member is variator, and that, if we introduce the new independent parameters converted from construction parameters of zoom system, the compensation equations of the both type systems become the same form theoretically. We also found that common equation can be applied for the zooming function of both focal and afocal systems. Starting with these basic equations; another classification may be obtained for mechanical compensation types, and it may be possible to position each zoom type in the succession.
For an application, we made the semi-automatic design program for optimizing the thin zoom lens system by this theory. In the last part of this paper, we showed two examples obtained by this program.