The Abbe Diffraction Theory. By J. W. Gordon. 389 



illumination, these fractions represent with sufficient accuracy for 

 present purposes the brightness of these several subsidiary maxima, 

 and it is evident that they fall off rapidly, and must therefore, save 

 with the very strongest illumination, become invisible at no great dis- 

 tance from the focus. In fact, with low illumination only the two 



nearest images, those having the maximum brightness of Q — , can 



be seen. With strong illumination the next pair will come into view, 

 while dazzling light, such as direct sunlight or the image of a flame, 

 may bring a much larger number into sight. It will, however, serve 

 our present purpose to discuss the case in which only the first pair of 

 images are conspicuous enough to be seen when isolated in the field 

 of the Microscope. 



A very little consideration will show that the shape of the 

 flanking image, being determined by the same factors and by almost 

 the same co-efficients as the shape of the central disc, will closely 

 resemble its primary ; and, without pausing over the mathematical 

 proof, I may perhaps at once premise that the parallelogram will 

 produce secondary images scarcely distinguishable in form from the 

 primary, and a circle will produce concentric rings. 



We have now reached the point at which these results may be 

 applied to the explanation of the phenomena of resolution in the 

 Microscope ; but in view of the Abbe experiments it will be of interest 

 to carry the investigation of the antipoint produced by the rectangular 

 aperture one step farther. The case so far considered is that of a 

 single opening, but the opening may be doubled, or it may be trebled, 

 as it is in the Abbe diaphragm of fig. 79. 



The superposition of the different patterns of antipoint produced 

 by the several members of the series of apertures under these 

 conditions of course gives complexity to the resulting figure; but 

 when the spaces between apertures have the same diameter as 

 the apertures themselves, the resulting antipoint is simple. For the 

 group of three apertures in that case yields an antipoint having 

 the same form and dimensions as would a rectangle having a diameter 

 equal to three times the diameter of any one of the apertures or spaces. 

 The distribution of light will, however, be different; for the dark 

 spaces, which only shut off from the first flanking image as much 

 liu r ht as would in any case be suppressed by interference, cut off from 

 the principal image § of the light which would otherwise reach it. 

 The relative brightness of the secondary image is therefore enhanced 

 in this proportion ; but obviously the increased relative value is secured 

 by loss of light to the focus, not by gain to the excentric image. In 

 this case, therefore, the secondary images are of much greater import- 

 ance to the appearance of the antipoint than when it is produced 

 by an uninterrupted aperture. Their importance may be still farther 

 increased by adding to the number of apertures, so that the limit 

 is reached only when the first flanking images become equally bright 



