An object such as Pleurosigma angulatum, which gives six diffraction spectra arranged as in [Fig. 33], should, according to this theory, show markings in a hexagonal arrangement. For there will be one set of lines at right angles to b, a, e, another set at right angles to c, a, f, and a third at right angles to g, a, d. These three sets of lines will obviously produce the appearance shown in [Fig. 33]a.
Fig. 33.
Fig. 33a.
Fig. 34.
A great variety of appearances may be produced with the same arrangement of spectra. Any two adjacent spectra with the central beam (as b, c, a) will form equilateral triangles and give hexagonal markings. Or by stopping off all but g, c, e (or b, d, f), we again have the spectra in the form of equilateral triangles; but as they are now further apart, the sides of the triangles in the two cases being as √3 : 1, the hexagons will be smaller and three times as numerous. Their sides will also be arranged at a different angle to those of the first set. The hexagons may be entirely obliterated by admitting only the spectra g, c, or g, f, or b, f, etc., when new lines will appear at right angles, or obliquely inclined, to the median line. By varying the combinations of the spectra, therefore, different figures of varying size and positions are produced, all of which cannot, of course, represent the true structure. Not only, however, may the appearance of particular structure be obliterated or created, but it may even be predicted before being seen under the microscope. If the position and relative intensity of the spectra in any particular case are given, the character of the resultant image, in some instances, may be worked out by mathematical calculations. A remarkable instance of such a prediction is to be found in the case recorded by Mr. Stephenson, where a mathematical student who had never seen a diatom, worked out the purely mathematical result of the interference of the six spectra b-g of [Fig. 33] (identical with P. angulatum), giving the drawing copied in [Fig. 34]. The special feature was the small markings between the hexagons, which had not, before this time, been noticed on P. angulatum. On more closely scrutinizing a valve, stopping out the central beam and allowing the six spectra only to pass, the small markings were found actually to exist, though they were so faint they had previously escaped observation until the result of the mathematical deduction had shown that they ought to be seen.
These experiments seem to show that diffraction plays a very essential part in the formation of microscopical images, since dissimilar structures give identical images when the differences of their diffractive effect is removed, and conversely similar structures may give dissimilar images when their diffractive images are made dissimilar. Whilst a purely dioptric image answers point for point to the object on the stage, and enables a safe inference to be drawn as to the actual nature of that object, the visible indications of minute structure in a microscopical image are not always or necessarily conformable to the real nature of the object examined, so that nothing more can safely be inferred from the image as presented to the eye, than the presence in the object of such structural peculiarities as will produce the particular diffraction phenomena on which these images depend.
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]