Originally Posted by A49

....Bob Atkins cites your "square formula" which is also written in the high res photography bible "Image Clarity" by Williams. Atkins doubt it. He writes here http://www.bobatkins.com/photography/technical/rrs.html :
[I]
This relationship is often given as:
1/Rs^2 = 1/Rf^2 + 1/Rl^2

where Rs is the final system resolution, Rf is the film resolution and Rl is the lens resolution. ... However it is not based on any sound theory and can, in fact, be shown to give erroneous results when used improperly. ...
Actually, the formula is a mathematically rigorous result under the conditions described in my post. It can be derived by a fairly simple derivation involving the moments of a distribution, though I don't recall the details of the derivation at the moment. It even applies to distributions that do not have a finite second moment, such as the Lorentzian function, provided one defines peak width in a slightly different way.

The only time it does not apply is if the system does not obey the stated conditions, mostly related to linear system response, the absence of statistical correlation between the lens and the film response functions, and the particulars of how one defines peak width or resolution. Other formulas may apply if the conditions are not satisfied, and in a more general case there may be no simple closed form equation, but numerical methods could be used to calculate the quantities in those cases.