The Hclmholtz Theory of the Microscope. By J. W. Gordon. 407 



apertures, therefore, its diffraction can be calculated by the known 

 rule for the diffraction of light from plane wave-fronts, and if the 

 entire beam is transmitted both backward and forward the diffrac- 

 tion pattern in each of the focal planes will be that produced by 

 focussing these beams of unfocussed light. At last, therefore, the 

 whole problem has been reduced to the problem already solved of 

 the diffraction caused by cutting down a beam of parallel light. 



We can, moreover, by the same considerations determine the 

 question of the dimensions of the diffraction pattern produced by 

 the diffraction of light from spherical wave-fronts. For it can 

 be shown — see fig. 94 — that the axis of a diffracted beam con- 

 taining only parallel rays would cut a plane perpendicular to the 

 optical axis of the instrument at a distance from the axis propor- 

 tional to the distance of that plane from the aperture, and deter- 

 mined by the equation 77 — tan 6 r v ; e = tan r ( . If now these 



oblique rays are brought to focus by a refracting system capable 

 of yielding fiat fields in the conjugate planes e. . ,€i and 77. . .77, 

 respectively, these oblique parallel rays must be brought to focus 

 at distances such that 77 = sin 6 r and e = sin 6 r e , for this is the 



condition of correct image formation in these focal planes.* 



This again is a result of capital importance, which, however, is 

 not very clearly brought out by Prof. Helmholtz. Throughout 

 the paper he speaks of diffraction fringes, a term appropriate 

 enough to describe the coloured margins formed by diffracted light 

 about the edges of shadows and beams of unfocussed light, but 

 little enough suggestive of the " false disc " formed by a perfectly 

 corrected lens as the image of a luminous point. When in June 

 of 1901 I had the honour of laying before this Society some criti- 

 cisms of the Abbe theory, I ventured to define an antipoint as the 

 correctly focussed image of a luminous point, and that definition 

 has been, as I gather, very generally accepted. May I now pre- 

 sume to define it a little more closely, and to point out to you that 

 the correctly focussed image of a luminous point is an image of a 

 certain diffraction fringe, which may easily be defined, but cannot 

 usually be seen. Let the following diagram (fig. 94) serve to illus- 

 trate this connection. 



A beam of parallel light passes the aperture, say, from right to 

 left, is received on a screen placed at e . . . e x , and thereon projects a 

 shadow image of the aperture, the central ray passing through the 

 point e. A diffracted beam is thrown off at an angle 6, the central 

 ray of which intersects the screen at e x , so that the axial distance 

 of €1 = tan 6 r. The beam is now reverted through the aperture, 

 supposed now to be filled by a lens having a flat focal field in the 

 plane 17 . . . ij^ and its principal focus at 77 in this plane distant by 

 r from c the centre of the aperture. Then at rj-^ will be formed an 



* See Appendix, Note I. 



