The Abbe Diffraction Theory. By J. W. Gordon. 387 



limit will be somewhere short of this minimum point, for the value of 



m will become extremely small before becoming = _, or infinitesimal. 

 n 1 



The exact boundary of the visible area cannot therefore be deter- 

 mined. Its position will depend upon the total light received by the 

 antipoint, and will lie nearer to p 2 with strong illumination than with 

 weak. For this reason definition and resolving power are impaired 

 by excessive brightness of the image— a point as to which I propose 

 to add an observation later on. In any case, however, the point p 2 

 will be a landmark beyond which the central disc of the antipoint 

 cannot extend. 



Into the question of illumination of the central disc it is not 

 necessary to enter more fully here, but we have still to consider its 

 form. This may of course be traced by laying down as a boundary 

 the line which forms the locus of the limiting point p 2 . The position 



of this point is determined, as we have seen, by the expression - d. 



a 



Now, for any given antipoint at any given time, X will have an 

 invariable value, and d a value subject only to such slight variations 

 as we may ignore. But a may vary to any extent, for it is a function 

 of the diameter of the aperture which determines the shape and 

 dimensions of the wave-front, and not necessarily a linear function. 

 [Moreover, it may vary in the same aperture for every several diameter. 

 Take, for instance, the cases of circular, square, and oblong apertures. 

 The circle is perfectly symmetrical, and has only one value for all its 

 diameters. The square is symmetrical along two axes, but has dif- 

 ferent values for its axial and diagonal diameters. The long rectangle 

 is bilaterally symmetrical along each of its principal axes separately, 

 but with very different values for the two sets of diameters. The 

 central disc of the antipoint produced by these several apertures will 

 vary correspondingly. The circle will of course produce a circle, since 

 the inverted symmetry can still only be satisfied by the circular form. 

 Here a is simply proportional to the diameter. The square in like 

 manner will produce a form approaching to the circle, but having 

 indentations opposite to the salient angles of the aperture. With 

 strong illumination these indentations become so pronounced as to 

 give to the antipoint a cruciform appearance. But the long rect- 

 angle will produce a very remarkable modification in the form of 

 the antipoint. The rule that the diameter of the antipoint must 

 vary inversely as the diameter of the aperture results in this case in 

 a very marked deviation from the circular form of the antipoint. 

 Clearly the elongated form of the aperture must result in a recipro- 

 cally compressed form of the antipoint — i.e. the antipoint will appear 

 elongated along the axis which is shortened in the aperture, com- 

 pressed along the axis which is the long axis of the aperture. Thus 

 the antipoint will present a general resemblance to the aperture with 

 its axes interchanged, and in the case of a much elongated aperture 



2 D 2 



