"The fact that one theory is consistent and the other is not does not necessarily mean that the former is more accurate than the latter."The footnote is in reference to the Kirchhoff diffraction integral which was derived under two inconsistent assumptions for the boundary conditions on the field. Despite these inconsistencies, the theory gives a very good prediction for the diffracted field far from a large aperture.
Kirchhoff's theory is also a good demonstration of the fact that mathematical consistency and exactness does not mean that a theory makes good predictions or can be used to calculate physical quantities. Experimentation must validate a theory's ability to do so.
