Further investigations and experiments led Abbe to discard so much of his theoretical conclusions relating to superimposed images having a distinct character as well as a different origin, and as to their capability of being separated and examined apart from each other. In a later paper he writes: “I no longer maintain in principle the distinction between the absorption image or direct dioptrical image and the diffraction image, nor do I hold that the microscopical image of an object consists of two superimposed images of different origin or a different mode of production. Thus it appears that both the absorption image and the diffraction image he held to be equally of diffraction origin; but while a lens of small aperture would give the former with facility, it would be powerless to reveal the latter, because of its limited capacity to gather in the strongly-deflected rays due to the excessively minute bodies the microscopical objective has to deal with.”[11]
Abbe’s theory of vision has been questioned by mathematicians, and since his death Lord Rayleigh went more deeply into the question of “the theory of the formation of optical images,” with special reference to the microscope and telescope. He has shown that two lines cannot be fairly resolved unless their components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the aperture; also, that the measure of resolution is only possible with a square aperture, or one bounded by straight lines, parallel to the lines resolved.
Lord Rayleigh’s Theory of the Formation of Optical Images, with Special Reference to the Microscope.[12]
Of the two methods adopted, that of Helmholtz’s consists in tracing the image representative of a mathematical point in the object, the point being regarded as self-luminous; that of Abbe’s the typical object was not, as we have seen, a luminous _point_, but a _grating_ illuminated by plane waves of light. In the latter method, Lord Rayleigh argues that the complete representation of the object requires the co-operation of all the spectra which are focussed in the principal focal plane of the objective; when only a few are present the representation is imperfect, and wholly fails when there is only one. He then proceeds to show, by the aid of diagrams and mathematical formula, how the resolving power can be adduced.