Highly Magnified Images. By J. W. Gordon. 



11 



From this result several very important inferences may be 



drawn. Thus, since the expression 



is independent of the 



2 sin u 

 dimensions of the aperture, we may infer that — 



VI. The dimensions of the antipoint depend only upon the 

 divergence angle of the focussed beam, and are in no way determined 

 by the magnitude of the aperture causing diffraction. This sounds 

 surprising, for we know that the sine of the diffraction angle is 

 inversely proportional to the diameter of the aperture, and are apt 

 to assume that the dimensions of the antipoint will vary in a 

 similar fashion. But a little reflection suggests that the divergence 

 angle must have more to do with the diameter of the antipoint than 

 has the breadth of the aperture, for, the divergence angle remain- 

 ing unchanged, the distance of the aperture must be proportional to 

 its diameter. Fig. 6 illustrates this relation. 



Here, whether we take the aperture to be A 1} A 2 or A 3 , it is 



Fig. U. 



plain that the sine of the diffraction angle multiplied by the dis- 

 tance of the aperture from the point rj will be a constant 



quantity = •— ? — ; in other words, the antipoint will be un- 

 2 sin u 



changed whatever change may take place in the length of the 



beam, provided that its divergence angle remains unchanged. 



From this principle Helmholtz deduces a very elegant result. 



Since the antipoint depends only upon the divergence angle of the 



focussed beam, it can make no difference to it, and no difference 



therefore to the state of resolution of the image at what point in 



the system the diaphragm is placed, by which the beam is defined 



and diffraction caused. Therefore the actual dimensions of the 



antipoint will be determined by that aperture which has the 



smallest optical projection upon the principal plane of the entire 



optical system. Furthermore, it is clear that there can only be one 



such aperture, and therefore, whether we consider the beam to be 



transmitted upward towards the ocular and out through the eye- 



