The ideal isothermal analysis is often called the Schmidt analysis after Gustav Schmidt (Sep 16,1826 - Jan 27, 1883) who published it in 1871 and provided a closed form solution for the case of sinusoidally varying volumes of the expansion and the compression space.
What distinguished the ideal isothermal analysis from others like the ideal adiabatic analysis is that it makes the assumption of perfect heat transfer from the working gas to/from the containing solid surfaces in the following sense :
In addition, the following two assumptions are made :
An immediate consequence of these assumptions, due to the law of conservation of energy, is that the cooler and heater space each have zero net heat transfer into/out of the working gas during a complete cycle. The task of heat transfer into the gas is provided by the expansion space and heat removal is conducted by the compression space. Except maybe for LTD Stirling engines (Low Temperature Differential) these spaces are clearly not designed for this purpose.